12
S. A. Majid and S. Tripathi
1 Introduction
Scouring is a very complex process that occurs due to the interaction of the flow with
the bed material. Different theories have been proposed to explain the inception of
sediment motion. The most widely accepted theory is the Shield’s theory [4]. It states
that the motion of sediment is initiated only when the bed shear stress exceeds the
critical shear stress. The numerical modelling for scouring by Shield’s theory would
thus require a good understanding of flow as well as bed properties. An extensive
literature exists on fluid mechanics as well as sediment transport and scouring [1, 5].
Scouring can occur due to a localised obstruction to the flow or due to contraction
in the flow passage. The contraction can be either vertical or horizontal. During flood
events, a pressure flow situation may be developed in the passage under the bridge.
The pressure flow is associated with increased bed stresses increasing the chances of
scouring in the passage [5]. The local scouring has been studied extensively from both
numerical and experimental perspectives. In comparison, the pressure flow scouring
due to the vertical contraction has received less attention. A few experimental studies
have reported the magnitude of vertical contraction scouring to be a function of flow
velocity, degree and length of contraction together with the bed properties. A proper
physical understanding of the contraction scouring is lacking due to the absence of
velocity field data. Further, no numerical approach has been reported to study the
pressure flow scouring due to vertical contraction.
2 Numerical Model
In the present study, clear-water pressure-flow scouring is simulated by solving the
Navier–Stokes equations coupled with k-E closure turbulence model and MeyerPeter and Müller bed load transport model [3]. Clear-water scouring is achieved by
setting the average approach flow velocity below the critical velocity for sediment
motion inception. The critical velocity is calculated by Neil’s formula:
V c = 1.52
g(s − 1)D 50 (
y
D 50
)
1
6
(1)
where ‘y’ is the depth of flow, ‘s’ is the specific gravity and ‘d 50 ’ is the median
diameter of the bed material. This numerical model solves the time-dependent
Reynolds-averaged Navier–Stokes equations:
∂U i
∂t
+ U j
∂U i
∂ x j
=
1
ρ
∂
∂ x j
−Pδ i j − u i u j
(2)
where ‘U i ’ is the time-averaged velocity field, ‘ρ’ is the density of the fluid, ‘P’ is
the piezometric pressure field, ‘δ ij ’ is the Dirac-delta operator and ‘−ρu i u j ’ is the
Reynolds stress tensor. The first term on the left side of Eq. 2 is the transient term
S. A. Majid and S. Tripathi
1 Introduction
Scouring is a very complex process that occurs due to the interaction of the flow with
the bed material. Different theories have been proposed to explain the inception of
sediment motion. The most widely accepted theory is the Shield’s theory [4]. It states
that the motion of sediment is initiated only when the bed shear stress exceeds the
critical shear stress. The numerical modelling for scouring by Shield’s theory would
thus require a good understanding of flow as well as bed properties. An extensive
literature exists on fluid mechanics as well as sediment transport and scouring [1, 5].
Scouring can occur due to a localised obstruction to the flow or due to contraction
in the flow passage. The contraction can be either vertical or horizontal. During flood
events, a pressure flow situation may be developed in the passage under the bridge.
The pressure flow is associated with increased bed stresses increasing the chances of
scouring in the passage [5]. The local scouring has been studied extensively from both
numerical and experimental perspectives. In comparison, the pressure flow scouring
due to the vertical contraction has received less attention. A few experimental studies
have reported the magnitude of vertical contraction scouring to be a function of flow
velocity, degree and length of contraction together with the bed properties. A proper
physical understanding of the contraction scouring is lacking due to the absence of
velocity field data. Further, no numerical approach has been reported to study the
pressure flow scouring due to vertical contraction.
2 Numerical Model
In the present study, clear-water pressure-flow scouring is simulated by solving the
Navier–Stokes equations coupled with k-E closure turbulence model and MeyerPeter and Müller bed load transport model [3]. Clear-water scouring is achieved by
setting the average approach flow velocity below the critical velocity for sediment
motion inception. The critical velocity is calculated by Neil’s formula:
V c = 1.52
g(s − 1)D 50 (
y
D 50
)
1
6
(1)
where ‘y’ is the depth of flow, ‘s’ is the specific gravity and ‘d 50 ’ is the median
diameter of the bed material. This numerical model solves the time-dependent
Reynolds-averaged Navier–Stokes equations:
∂U i
∂t
+ U j
∂U i
∂ x j
=
1
ρ
∂
∂ x j
−Pδ i j − u i u j
(2)
where ‘U i ’ is the time-averaged velocity field, ‘ρ’ is the density of the fluid, ‘P’ is
the piezometric pressure field, ‘δ ij ’ is the Dirac-delta operator and ‘−ρu i u j ’ is the
Reynolds stress tensor. The first term on the left side of Eq. 2 is the transient term