(1)
Yongsheng
Wu, Zhaoyin Wang
Department of Hydraulic Engineering, Tsinghua University, 100084, Beijing, China
Jijian
Lian, Qinghe Zhang
School of civil Engineering, Tianjin University, 300072, Tianjin, China
Abstract:
The concentration gradient of sediment often occurs in estuarine and coastal
regions. The velocity profile of current with waves is affected by the gradient
and exhibits different forms. This paper presents a k-εturbulent
model for wave-current combined flows of the full depth of the estuarine and
coastal region simulating the
effect of sediment concentration on the velocity profile. A quantitative
analysis shows that the existence of concentration gradient decreases the effect
of waves on the mean current velocity in the case of following current.
Keywords: stratification effect, interaction of wave-current, mean velocity profiles
The calculation of mean velocity is an important part of sediment transport and pollution diffusion models. Stratified flows occur often in estuaries and coastal regions due to suspended sediment. The calculation of mean velocity should be performed by taking the currents, waves and the stratification into account. The velocity profile is different under different conditions of waves and stratifications. The mechanism of the profile variation was studied with flume experiments and numerical models without considering the effect of sediment.
Malarkey et al.(1998), Nielsen and You (1996) and Groeneweg and Klopman (1998) studied the interaction of waves and currents with mathematical models and physical experiments. For example, many experiments show that the velocity profile of current following non-breaking wave is more uniformly distributed than that of the current without waves. In the case of the current opposing waves a more straight profile is observed. It is more complicated that the combination of wave and currents cause intensive erosion and resuspension of bottom sediment, generating a vertical stratification, which in turn affects the flow structures. Nevertheless, there are few studies, neither mathematical nor the experimental, about the change of the velocity profiles under the existence of suspended sediment. For the sake of developing a model of mean velocity under wave-current combination conditions, the mechanism of velocity profile affected by sediment stratification and wave-current interaction must be studied.
The interaction between sediment and flow is complicated. The effects of sediment on the flow structure have been studied from laboratory experiments and field observations (Mehta,1989a, 1989b; Parker,1997, Wang and Larsen,1998). Although the effect of sediment stratification on the flow structure is qualitatively understood, a model is yet to be developed for the flows under the wave-current combined conditions. Two methods were used to consider such effect in the literatures, in which the turbulent water flow is explicitly coupled with the sediment movement. One kind of the model employs the concept of Richardson number Ri and the eddy viscosity is a function of Ri. The second considers the buoyancy effect of sediment through a sediment induced buoyancy term in the turbulence closure equation. Le Hir et al. (1997,1998) and Lang et al. (1989) and Wright et al. (1999) modeled the concentration stratification with Richardson number, through concerting parameters with data. Such models can not be used for other flow conditions. More sophisticated model is needed with comprehensive turbulent equations., Sheng (1986,1989) developed a Reynolds stress model and a relative simple TKE (turbulent kinetic energy) model. These models mainly focused on the change of turbulence intensity within boundary layer, and the numerical results agree with the experimental results. Recently, Winterwerp (1999) developed the k-εturbulent model to estimate the buoyant effect of concentration stratification. In his paper, many physical processes relevant to cohesive sediment transport are considered including the buoyancy effect. But the wave motion is not considered. This model is validated through application to well-documented laboratory experiments and field measurement. Although the Reynolds stress model is found to give more accurate results, the k-εturbulent model is simpler and more applicable. This paper presents a k-εmodel studying the effect of concentration stratification and its effect on the velocity profiles. The simulation region is from bed to the free surface, within and out of the boundary layer. The water-sediment mixture is treated as a one phase flow in which all particles follow the turbulent movement, but for their settling velocity (Winterwerp, 1999; Le Hir, et al. 1998; markafsky, et al. 1989).
For imcompressible stratified fluid, Navier-Stokes equation for a two-dimensional problem reads:
(1)
(2)
and the continuity equation reads:
(3)
where
,
,
are horizontal and vertical velocities and
pressure respectively,
is the kinematic viscosity of water-sediment mixed body,
is the density of water-sediment mixed body,
is the gravity acceleration,
is the time, and
,
are horizontal and vertical
coordinate respectively.
The distribution of suspended sediment concentration in water column is governed to good approximation by the following advection-diffusion equation:
(4)
where c
is sediment mass concentration,
s is eddy diffusivity, ws is effective settling velocity for sediment
suspension.
The detailed derivation of wave and current motion equations can be found in Wu, et. al. (2000), and the final equations of wave and current equations are given only.
For wave motion equations, after involving the boundary conditions, the finial form is given as:
(5)
(6)
(7)
where
. K is complex wave number,
,
is the real part, and
is the wave damping coefficient,σis the wave
frequency,
and
are velocity amplitudes of the periodical components of the instantaneous
velocities
and
, and i is imaginary unit,
.
For current motion, the integration form is given as:
(8)
(9)
(10)
where d is
water depth,
is time-independent surface elevation,
is time-independent pressure, and z0is theoretical bottom elevation,
where z0 is theoretical bottom elevation, for a rough surface,
(
is Nikuradse’s equivalent roughness), and
is used for a smooth surface (
is wave maximum friction velocity).
The calculation of terms of
,
is given as follows:
(11)
(12)
where the subscript i and r denote the real part and imaginary part respectively.
As is known to all, eddy viscosity can be modeled by following relationship:
(13)
where
denotes
or
,
and
are the turbulent kinematic energy and dissipation rate respectively, The
governing equations of
and
are given as:
(14)
(15)
The model constants are given by the standard setting:
(16)
In equation (4), the eddy diffusivity
for suspended sediment,
s is assumed proportional
to the eddy viscosity,
, for the turbulent momentum transport:
(17)
where the
proportional coefficient
T is well known as the turbulent Prandtl-Schmidt number. However, the
value of
T is less well established. In present paper, we follow Winterwerp
(1999),
T=0.7. Moreover, for stable
stratified flow,
3c in equation (15) is recommended to be 1.0 according to the
discussion in Winterwert (1999).
The boundary conditions of wave and current motion equations consist of no slip conditions and the free surface conditions, the detail expressions can be found in Wu et, al. (2000).
The boundary conditions for k-e model can be described as:
(18)
(19)
(20)
(21)
where κis von karman constant, and it is equal to 0.4.
For the sediment advection-diffusion equation, for the case of no sediment exchange between bed and water, the boundary conditions are given as:
at bottom
(22)
at free surface
(23)
From the above derivation, it shows that the
wave, current and suspended cohesive sediment are coupled each other. To
validate the model, following two steps are used. (1)
the model is validated with clear water experimental data, (2) the
experimental data in steady current are employed to validate the suspended
sediment effect of the model.
The measurements of Klopman(1994)
were taken in a so called Scheldt flume, and the wave height is 0.12m, water
depth is 0.5m, wave period is 1.44s, the reference elevation is o.4m, the
reference velocity is 0.12m/s for following current and 0.23m/s for opposing
current, and Nikradse’s roughness is 1.2mm. The result of comparison is given
in Fig. 1. It is shown that the prediction of present model on the experimental
data is fairly good.
Fig. 1 The comparison of the experimental data of Klopman(1994) with the present model
In present paper, the effect of suspended cohesive sediment on flow is validated by comparisons with Coleman (1981) experimental data. Because wave was absent in Coleman (1981), current is only considered in this validation process. The experimental conditions are as follows: depth-mean velocity is 0.96m/s, water depth is 0.172m, the homogeneous concentration, cmean, is 6.0g/l, sediment median diameter is 105μm, bed is hydraulic smooth. The comparison results are given in Figure2-figure3.
Fig. 2 The comparison with Coleman’s data Fig. 3 The comparison with Coleman’s data
From the comparisons, we can conclude that numerical model developed in this paper is able to reproduce the effects of the suspended sediment on vertical velocity profile and on sediment mixing process.
The procedure of modeling simulation is as follows, without taking into account the effect of suspended sediment, the initial conditions, such as water depth, depth-mean current velocity, wave height, wave period, bed condition and homogeneous mean concentration profile are imposed, i.e. decoupled calculation. After convergence, the effect of suspended sediment is involved from the previous calculation results. Considering the effect of sediment concentration on the settling velocity, hindered velocity formula is employed. In this paper, the settling velocity profile used in Le Hir (1998) is used. Moreover, the kinetics viscosity formula of Mehta(1989a) is also used for considering the molecular viscosity increase of the cohesive sediment-water mixture.
The conditions of numerical application here are as follows, water depth is 0.5m, wave height is 0.16m, wave period ia 2.0s, bed roughness is 1.0cm, the depth-mean current velocity is 0.1m/s, and the interacting angle of wave and current is 0°. The numerical results are presented in Fig.4-Fig.5. The solid lines are the numerical results without sand the dot lines are the results with sand. In the fig.4, the effects of sand of the mean velocity cause the mean velocity near bed decrease and the mean velocity at outer layer increase. It is noticeable that the influence of suspended sediment can not be omitted for a correct prediction of the mean current profiles. Fig.5 is the comparisons of Reynolds shear stress under the cases of decoupled and coupled results. The profiles of both two are very similar, however, the results of coupled results are smaller than that of decoupled
Fig. 4 Comparison of coupled and decoupled Fig. 5 The comparison of coupled and
numerical decoupled numerical simulations of Reynolds stress
simulations of mean velocity
A turbulent kinetic energy closure model was
developed, involving buoyancy effect induced by density stratification. In clear
water, the mechanism of change of mean velocity distribution within and outside
boundary layer is attributed to the balance the contribution of wave to current
and mean surface gradient. However, in the sediment-water mixed flow, the
turbulence damping induced by buoyancy effect of suspended sediment is very
obvious, which affects the mean velocity profile significantly. Under the
wave-current-suspended sediment combined flow, the change mechanism of mean
current velocity is not only related to the strength of waves, but also related
to the suspended sediment properties, such as concentration profiles and its
physical-chemical characters. For the following current, the existence of
suspended sediment decreases the effect of waves on current and leads the
velocity profile tend to the clear and pure current. For the opposing current,
the existence of suspended sediment increases the effect of waves on current.
References
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