Longitudinal dispersion of heavy particles in an oscillating tunnel and application to wave boundary layers
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Abstract
The present research aims at getting an understanding of the process of dispersion of surface sediment in an oscillatory boundary layer, which may represent an idealised case of, for example, a stockpile area where excavated sediment is stockpiled temporarily (or permanently). The process is studied numerically, using a randomwalk particle model with the input data for the mean and turbulence characteristics of the wave boundary layer picked up from a transitional twoequation k–\(\omega \) Reynolds averaged Navier–Stokes model and plugged in the randomwalk model. First, the flow model is validated against experimental data in the literature. Then, the randomwalk dispersion model is run for different oscillatory flow cases and for a number of steady flow cases for comparison. The primary sediment grains of concern are fine sediments (with low fall velocity), which would stay in suspension for most of the time. Nevertheless, the dispersion of neutrally buoyant and heavier particles that spend most of their time in close vicinity to the bed are also discussed. The numerical model results are compared with the results of a series of experiments carried out in an oscillating Utunnel facility. The results are found to be in general agreement both qualitatively and quantitatively. In the last part of the study, an example application of the present model for fine sand dispersing in a wave boundary layer under storm conditions is given.
Keywords
Longitudinal dispersion Suspended sediment Oscillatory boundary layer Randomwalk model RANS modelling Utunnel experimentsList of symbols
 a
Amplitude of oscillatory motion
 \(a_\mathrm{r}\)
Standard normal variable
 \(\alpha _{\omega }\), \(\beta _{\omega }\), \(\sigma _{\omega }\), \(\alpha ^{*}\), \(\beta ^{*}\), \(\sigma ^{*}\), \(\alpha _{0}^{*}\), \(\alpha _{0}\), \(\beta _{0}^{*}\), \(\sigma \)\(_\mathrm{d}\), \(\sigma \)\(_\mathrm{do}\), \(\sigma \)\(_\mathrm{do}\), \(R_{\beta }\), \(R_{\beta }\), \(R_\mathrm{k}\), \(S_\mathrm{R}\), \( K_\mathrm{r}\)
Constants and variables used in k–\(\omega \) turbulence closure model
 b
Dispersion model tuning parameter
 \(\beta \)
Rouse parameter
 c
Concentration of particles (number of particles per unit area)
 C
Nondimensional concentration of particles (probability density function)
 \(C_{0}\)
Zeroth moment of concentration
 d
Diameter of particle
 \(\delta \)
Thickness of oscillatory boundary layer
 \(D_{1}\)
Dispersion coefficient
 \(\delta _\mathrm{b}\)
Thickness of viscous sublayer
 \(\Delta t\)
Duration of particle vertical displacement
 \(\Delta y\)
Vertical displacement of a particle
 \(\varepsilon \)
Dissipation rate
 g
Gravitational acceleration
 h
Oscillating tunnel halfthickness or water depth
 \(H_\mathrm{rms}\)
Rootmean square wave height
 k
Turbulence kinetic energy
 \(\kappa \)
Von Kármán constant
 \(k_\mathrm{s}\)
Nikuradse’s equivalent sand roughness
 \(k_\mathrm{s}^{+}\)
Roughness Reynolds number
 \(\ell \)
Model turbulence length scale
 \(\ell _\mathrm{p}\)
Length scale of particle vertical motion
 M
Integer of model resolution
 \(\nu \)
Kinematic viscosity
 N
Number of particles
 \(\nu _\mathrm{T}\)
Turbulence viscosity
 p
Pressure
 \(P_{x}\)
Pressure gradient term
 \(\theta \)
Shields parameter
 \(\rho \)
Specific mass
 \(Re_\mathrm{T}\)
Model Reynolds number
 \(Re_\mathrm{w}\)
Wave Reynolds number
 S
Autocorrelation coefficient
 s
Specific gravity of particle
 t
Time
 T
Nondimensional time
 \(t_{0}\)
Initial time
 \(t_\mathrm{end}\)
Final time
 \(t_\mathrm{storm}\)
Duration of storm
 \(T_\mathrm{w}\)
Period of oscillatory motion
 \(T_\mathrm{z}\)
Zeroupcrossing period of waves
 U
Particle velocity
 \(\overline{U} \)
Ensembleaveraged particle velocity
 \(\bar{u}\)
Reynoldsaveraged Eulerian flow velocity
 \(u'\), \(v'\), \(w'\)
Fluctuating velocity component along x, y and z axes, respectively
 \(U_{0}\)
Free stream velocity
 \(U_\mathrm{f}\)
Friction velocity
 \(U_\mathrm{fm}\)
Amplitude of friction velocity
 \(U_\mathrm{m}\)
Amplitude of free stream velocity
 \(\omega \)
Specific dissipation rate
 \(w_\mathrm{s}\)
Particle settling velocity
 \(\omega _\mathrm{w}\)
Angular frequency of oscillatory flow and waves
 X
Particle position
 \(\overline{X} \)
Ensembleaveraged particle position
 \(X_{0}\)
Nondimensional free stream displacement
 \(\xi \)
Nondimensional horizontal distance
 x
Horizontal distance
 y
Vertical distance measured from the bed
 Y
Nondimensional vertical distance
 \(y^{+}\)
Vertical distance in wall units
 \(y_{2}\)
New vertical position of a particle
 \(y_{n0}\)
Initial vertical position of the \(n\mathrm{th}\) particle
1 Introduction
Construction operations in the marine environment may require vast amount of dredging by means of different methods. During the handling and disposal process of excavated (dredged) sediment, various amounts of this rather fine material will be spread into the marine environment. Eventually, the dredged material is either used as fill material for other parts of the construction or is disposed to a stockpile area in the sea. Since the marine environment, especially the nearbed region, is highly dynamic in terms of current and/or wave action, the disposed sediments will go through a series of transport and mixing processes (i.e. advection, diffusion, dispersion). It is of great importance to understand the mechanics of these processes for planning and managing of marine dredging operations.
Dispersion in bottom boundary layers is the most dominant of all these mixing processes qualitatively and quantitatively. In this process, the joint action of shear (mean velocity gradient) and wallnormal turbulence fluctuations in the boundary layer near the bed disperses the sediment particles horizontally. The dispersion coefficient is the measure for the rate of dispersion.
Longitudinal dispersion in steady flows has, in the past, drawn quite a lot of attention, with the pioneering work of Taylor (1953, 1954), followed by Aris (1956), Elder (1959), Fischer (1966, 1967), Sayre (1968), Chatwin (1973), Sumer (1973, 1974), Pedersen (1977), Fischer et al. (1979), Chatwin and Sullivan (1982), Allen (1982), Smith (1983) and Demuren and Rodi (1986) among others. The findings of these studies were often applied to channels, transmission pipes, atmospheric boundary layers, rivers, tidal inlets and estuaries.
Wave boundary layers, on the other hand, are formed over the seabed under waves. They are characterised by a very small vertical extent (in the order of magnitude of 20–30 cm at most), with very strong shear and turbulence, two important “ingredients” for dispersion. Unlike the steady flow situations, there are not many studies dealing with dispersion processes in unsteady flows, particularly in oscillatory and wave boundarylayer flows. Aris (1960) was the first to deal with the dispersion of a solute in periodically altering flows, albeit for the laminar case. Chatwin (1975) investigated the longitudinal dispersion of a passive neutrally buoyant dispersant in oscillatory pipe flow. Smith (1982), Yasuda (1982) and Yasuda (1984) studied the dispersion process in oscillatory twodimensional boundary layers, which resembles tidal flow boundary layers rather than wave boundary layers. Yasuda (1989) extended his previous research to dispersion of suspended (heavy) particles. Mei and Chian (1994) conducted a theoretical study in which they paid special attention to the dispersion of small suspended particles in wave boundary layers, including the convection due to the steady streaming. These works all used a timeinvariant eddy viscosity when describing the turbulent flow. Ng (2004) studied the dispersion process in oscillatory boundary layers using a timedependent turbulent diffusivity and found that the time dependency of the eddy viscosity could not be neglected since it generated a significant difference in terms of dispersion coefficients. Recently, Mazumder and Paul (2012) carried out a numerical study in which they modelled the dispersion of suspended sediments in oscillatory boundary layers including the settlement, temporary storage and reentrainment processes.
The present research aims at getting an understanding of the process of dispersion of the surface sediment in an oscillatory boundary layer, which may represent an idealised case of, for example, a stockpile area where excavated sediment (stockpiled temporarily or permanently) is subject to waves. Section 2 presents a singleparticle analysis for the longitudinal dispersion, first for a steady turbulent flow in an open channel, followed by an extension of the latter analysis to an oscillatory turbulent flow in an oscillating tunnel. Section 3 presents the numerical model adopted in the study, comprising a k–\(\omega \) model to calculate the “background” flow, and a randomwalk model to calculate the longitudinal dispersion. Section 4 describes the singleparticle experiments conducted in an oscillating tunnel. Section 5 presents the results including the validation of the numerical model, and comparison with the experimental data obtained in the experimental campaign. The paper ends with a final section, Sect. 6, in which the numerical model developed in the present study is applied to a reallife problem with dispersion of disposed sediment in a wave boundary layer under storm conditions. The present results and the existing knowledge (yet, highly limited) form a complementary source of information on the dispersion of sediment in wave/oscillatory boundary layers.
2 Longitudinal dispersion in an oscillating tunnel
In this section, the description of the longitudinal dispersion process will be discussed from the point of view of oneparticle analysis. First, the dispersion process will be discussed for an open channel flow (Sect. 2.1) and, subsequently, the analysis will be extended to the idealised case of dispersion in an oscillating tunnel (Sect. 2.2).
2.1 Longitudinal dispersion in an open channel flow
Because of the presence of the free surface and the bottom, and more importantly the conditions ensuring that the particle remains in the flow, the horizontal velocity of the particle is necessarily a stationary random function of time as soon as the influence of the special choice of the point on the crosssection where the particle was released has been lost.
Sumer (1974) developed a statistical formulation to describe the longitudinal dispersion of heavy particles in such a flow environment. In what follows is a summary of this formulation, which will be extended to the case of oscillatory flow in an oscillating tunnel in the next section.
2.2 Longitudinal dispersion in an oscillating tunnel
Similar to the open channel case, we will describe the process of longitudinal dispersion of sediment particles in the oscillating tunnel, using the oneparticle analysis. We release the particle from a point in the tunnel (Fig. 2). The particle will travel under the combined action of the phaseresolved mean shearing motion, phaseresolved turbulence and gravity. Here, too, we consider that the flow conditions and the particle properties are such that the particle is maintained in the main body of the flow.
Because of the presence of the two walls of the tunnel, and more importantly the conditions ensuring that the particle remains in the flow, the horizontal velocity of the particle will be a stationary random function of time as soon as the influence of the special choice of the point on the crosssection where the particle was released has been lost, similar to the openchannel case. Therefore, the statistical formulation developed for the openchannel flow case should be equally valid for the oscillating tunnel case as well (Eqs. (1)–(4)), with \(\overline{U} \) and \(\overline{X} (t)\) in Eq. (1) identically equal to zero due to the symmetric periodic motion in the tunnel.
The nondimensional expression for the longitudinal dispersion coefficient, \(D_1 \) in Eq. (5), may be adopted for the oscillating tunnel case, replacing (1) the flow depth h (which is actually the thickness of the boundary layer in the open channel) with the thickness of the oscillatory boundary layer in the oscillating tunnel, \(\delta \), and (2) the friction velocity \(U_\mathrm{f}\) with the maximum value of the friction velocity associated with the oscillatory boundary layer in the oscillating tunnel, \(U_\mathrm{fm} \). Here, \(\delta \) is defined as the location of the point of maximum velocity from the bed measured at the phase \(\omega _\mathrm{w}t=90^\mathrm{o}\) (i.e. when \(U_{0}=U_\mathrm{m})\) (Jensen et al. 1989).
3 Numerical model
The numerical model has two components: (1) the flow model that will yield the phasedependent velocity and turbulent kinetic energy profiles, and (2) the randomwalk dispersion model that will allow the simulation of the dispersion process, based on the tracking of a single particle for multiple times with the input from the flow model. These components are explained below.
3.1 The flow model
Twoequation turbulence closure models for Reynolds averaged Navier–Stokes (RANS) equation have been widely used in fluid mechanic and coastal engineering for solving various problems related to turbulent boundary layers. These models either use the k–\(\omega \) or k–\(\varepsilon \) turbulence closure. In these models, it is possible to solve the Reynolds averaged velocities (i.e. mean velocities) along with wall shear stresses and turbulence kinetic energy on a timeaveraged or phaseresolved basis.
On the upper boundary (namely, at the centreline of the tunnel), a frictionless lid is considered, whereby the shear stress and vertical derivatives of all variables are set to zero.
When the flow model was run for steady flow conditions, the model time was run for long enough such that all the flow parameters would remain unchanged with time. When oscillatory flow conditions are simulated by flow model, it was seen that after 4\(\mathrm{th}\) or 5\(\mathrm{th}\) wave period the model output started to repeat itself periodically. As such, the flow model was run for a warm up time of 10 wave periods and the values in the final wave period of the model were extracted for repeated use in the dispersion model.
In this version of the model, secondary (convective) terms were not included in the model equations. Therefore, when simulating the oscillatory flow, the steady streaming associated with the wave boundary layers is not considered. For other details of the MatRANS model, Fuhrman et al. (2013) can be consulted.
3.2 The dispersion model
The dispersion process is studied, using a randomwalk model. The randomwalk model is a particlebased model where a number of particles (N) are released from a point source (or a line or plane source) one after another, each being tracked from time \(t=0\) to a time of \(t=t\). A particle placed at a vertical position is convected in the vertical direction by (1) the vertical fluctuating velocity, and (2) the fall velocity of the particle itself, while it is convected in the horizontal direction by the horizontal mean velocity. The model assumes that the vertical fluctuating velocity exists for a small time interval. The magnitude and direction of the vertical fluctuating velocity are the random elements of the model. The path of an initially marked particle position is calculated as the sum of a series of such small time intervals as the particle migrates through the statistical field variables. Many such paths are used to compile the statistical properties by means of an ensemble average. In the simulations, the particles are maintained in the main body of the flow all the time. To this end, the particles coming very close to the bottom are reentrained into the flow, as will be detailed later.
We note that no interaction between the particles (graintograin collision) is taken into account in the simulations, implying that simulations are not valid for extremely high concentrations. Also, the bouncing of particles at the bed is considered to be due to nearbed turbulence alone. This is a valid approximation for sediment covering silt, fine sand and even partly mediumsize sediment.
To the authors’ knowledge, Sullivan (1972) and Sumer (1973) were the first to use a randomwalk model for the purpose of simulating the longitudinal dispersion processes in a turbulent openchannel flow, Sullivan (1972) for neutrally buoyant particles, and Sumer for heavy particles. Bayazit (1972) also used a similar randomwalk model to calculate the settling distances of heavy particles in a turbulent openchannel flow.
In the present study, the simulations were carried out for three kinds of flow environments: (1) for a steady flow in a tunnel; (2) for an oscillatory flow in an oscillating tunnel, and (3) for an oscillatory flow in a wave boundary layer. The steady flow case is included as a reference case, which also enables the model validation as well. The model is also validated in the case of the oscillating tunnel against the laboratory experiments carried out in an oscillating tunnel, described in the next section. The third case, the wave boundary layer, is included to demonstrate the implementation of the model in a reallife situation.
The flow domain is defined as a 2D tunnel (Fig. 2). The hydrodynamic model is run for the halfspace 0 Open image in new window first, and then the flow domain for dispersion model is defined by vertically mirroring the hydrodynamic model output about the center line.
In the oscillatory flow cases, the value of \(\Delta t\) needs to be limited with a fraction of the wave period, \(T_\mathrm{w}\), considering that the timescale of turbulent motion should be significantly smaller than the wave period. In other words, the vertical travel of the particle cannot be sustained with a constant vertical velocity for long durations comparable with the wave period, given that the statistical properties of the turbulence characteristics are a function of the phase (i.e. \(\omega _\mathrm{w}t)\). For example, for each \(T_\mathrm{w}\)/4 time interval, the free stream velocity rises from minimum (0) to maximum (\(U_\mathrm{m})\) along with the bed shear stress and associated turbulence characteristics. Thus, one would expect that \(\Delta t\) cannot exceed, say, a quarter of the wave period. The comparison of the numerical model results with that of the experiments showed that limiting the travel time to be \(\Delta t \le \) 0.15 \(T_\mathrm{w}\) gives a better match with the experiments. We further note that the difference in the dispersion coefficient caused by the selection of limiting value as 0.15 \(T_\mathrm{w}\) and 0.25 \(T_\mathrm{w}\) is rather small. It should also be mentioned that when \(\frac{a}{h}\ge 20\), the effect of limiting \(\Delta t\) on dispersion coefficient vanishes.
 1.
Run the flow model for the designated flow conditions and obtain \(\bar{u}( {\omega _\mathrm{w} \,t,y})\), \(k( {\omega _\mathrm{w} \,t,y})\) and \(\ell ( {\omega _\mathrm{w} \,t,y})\) for the lower half of the oscillating tunnel
 2.
Calculate \(\sqrt{\overline{v'^2} } ( {\omega _\mathrm{w} \,t,y})\) and \(\ell _\mathrm{p} ( {\omega _\mathrm{w} \,t,y})\) matrices through Eqs. (17) and (25), respectively
 3.
Mirror the matrices in step 2, to include also the upper half of the tunnel
 4.
Lineup N particles with the designated \(\beta \) at \(x=0\) and \(t=0\) along the vertical with intervals described by Eq. (23)
 5.Release the N particles one by one and run the dispersion simulation for each particle until \(t=t_\mathrm{end}\).
 (a)
For the given phase (\(\omega _{w }t)\) and vertical location (y) of the particle, pick up the values of \(\ell _\mathrm{p} \) and \(\sqrt{\overline{v'^2} } \) from the relevant matrices and calculate \(\Delta t\) via Eq. (24)
 (b)
Check if \(\Delta t \le \) 0.15\(T_\mathrm{w}\)
 (c)
 (d)
Calculate \(y_{2}\) from Eq. (28)
 (e)
Calculate \(\Delta x\) from Eq. (31)
 (f)
Update t, x and y as \(t =t +\Delta t\), \(x =x +\Delta x\) and \(y =y_{2}\)
 (g)
If \(t < t_\mathrm{end}\), continue the simulation with repeating the steps a–f
 (h)
If \(t > t_\mathrm{end}\), discard the last vertical travel of the particle and arrange a final travel by designating \(\Delta t=t_\mathrm{end}\) – t.
 (i)
Save the path of the particle and go to step 5 for a new particle
 (a)
 6.
After the last (\(N\mathrm{th}\)) particle completes its path, stop the dispersion simulation and calculate the relevant particle statistics.
4 Experiments
A spherical particle, printed in plastic using a 3D printer, was used in the experiments. The particle size, the specific gravity and the settling velocity of the particles were \(d=2.9\) mm, \(s=1.19\) and \(w_\mathrm{s}=9.8\) cm/s, respectively.
The free stream velocity was measured by a Dantec Dynamics FiberFlow laser Doppler anemometer (LDA) at 14.5 cm away from the bed (i.e. at the centreline of the tunnel). The LDA signal also served as a reference signal for monitoring the phase of the flow during particle tracking experiments. In addition to free stream velocity, velocity profiles were determined by means of LDA measurements conducted at 30 positions across the half thickness of the oscillating tunnel. At each location, velocity measurements were taken for 50 wave periods and ensemble averaging was conducted to obtain the Reynolds averaged velocity as a function of the phase, \(\bar{u}({\omega _\mathrm{w} t,y})\). These measurements are not included here for reasons of space.
For the determination of the friction velocity, \(U_\mathrm{fm}\), direct measurements of bed shear stress were conducted with a Dantec Dynamics 55R46 Hotfilm probe. These measurements were also conducted for 50 wave periods to obtain the ensembleaveraged values.
The experiments conducted in the oscillating tunnel
\(U_\mathrm{m}\) (m/s)  \(a=\frac{U_\mathrm{m} T_\mathrm{w} }{2\pi }\) (m)  \(Re_\mathrm{w} =\frac{U_\mathrm{m} \,a}{\nu }\)  \(U_\mathrm{fm}\) (cm/s)  \(\delta \) (cm)  a/h  \(\beta \)  \(\theta =\frac{U_\mathrm{fm} ^2}{( {s1})gd}\)  N  

Min.  Max.  Mean  
0.63  0.97  6.0 \(\times \) 10\(^{5}\)  3.3  1.5  6.7  7.2  0.22  119  534  286 
1.03  1.59  1.5 \(\times \) 10\(^{6}\)  4.9  2.8  10.7  4.8  0.47  356  630  511 
1.54  2.38  3.3 \(\times \) 10\(^{6}\)  6.6  3.9  16.4  3.6  0.83  140  350  240 
In the experiments, the particle was released into the flow domain multiple times (Table 1, last column). Then, the particle trajectory was videotaped, and subsequently the mean particle position and the variance along with the longitudinal dispersion coefficient were calculated. Similar particle tracking experiments were previously conducted by Sumer and Oguz (1978) and Sumer and Deigaard (1981) in an open channel flow.
One of the limitations of the experimental setup was that, it was not possible to track the particles too long before they disappear out of the light sheet. Typical durations of monitoring were about one wave period. It should also be noted that the particles might be lost for some frames in the video, but then found again and subsequently tracked. The last three columns of Table 1 refer to the minimum, maximum and mean number of particles that could be tracked at each frame during one complete wave period. Further details about the experiments can be found in Steffensen (2014).
5 Results
5.1 Validation of the flow model
The flow model used in this study, MatRANS, will be validated against different steady and unsteady flow cases (Fuhrman et al. 2013). For validating the results of the present modified version of the model, it is compared with the oscillatory flow measurements of the literature.
Similar to the flow model used in this study, recently Tang and Lin (2015) developed a RANS model established on the k–\(\omega \) baseline formulation, which can simulate the turbulent as well as transitional boundary layer flows. They also compared their findings with the experimental data of Jensen et al. (1989), yielding a good agreement.
5.2 Longitudinal dispersion in steady flow
The flow parameters used in the simulations are as follows: The half thickness of the tunnel, \(h=14.5\) cm, the mean flow velocity, V = 2 m/s, the friction velocity, \(U_\mathrm{f}\) = 8 cm/s. The simulations were conducted over a time period of \(0\le T\le 20\), which corresponds to \(0\le t\le 11\) s with the present flow parameters. The vertical and horizontal particle positions (snapshot of particle locations) were recorded by interpolation at frequent time intervals (with 0.1 in terms of nondimensional time). The simulations were conducted for different settling velocities, covering \(0\le \beta \le 1\).
The simulation results summarised in the preceding appear to be in good agreement with the existing information. This has provided confidence in the use of the present methods for the oscillatory flow cases studied in the present research.
5.3 Longitudinal dispersion in oscillating tunnel
The summary of oscillatingtunnel dispersion simulations
Group No.  Number of simulations  \(\beta \)  \(T_\mathrm{w}\) (s)  \(U_\mathrm{m}\) (m/s)  \(a=\frac{U_\mathrm{m} T_\mathrm{w} }{2\pi }\) (m)  Re \(_\mathrm{w}\)  h (cm)  a/h  \(U_\mathrm{fm}\) (cm/s)  \(\delta \) (cm) 

1  14  0.0  9.72  0.42–8.00  0.65–12.38  2.73\(\times \)10\(^{5}\)9.9\(\times \)10\(^{7}\)  14.5  4.5–85.4  2.7–26.4  0.6–14 
2  24  0.0  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–220  1.4–123.8  7.9–8.7  2.5–4.8 
3  24  0.1  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–220  1.4–123.8  7.9–8.7  2.5–4.8 
4  24  0.2  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–220  1.4–123.8  7.9–8.7  2.5–4.8 
5  24  0.3  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–220  1.4–123.8  7.9–8.7  2.5–4.8 
6  24  0.4  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–220  1.4–123.8  7.9–8.7  2.5–4.8 
7  24  0.5  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–220  1.4–123.8  7.9–8.7  2.5–4.8 
8  23  0.6  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–100  3.1–123.8  7.9–8.7  2.5–4.8 
9  10  0.8  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–56  5.5–123.8  7.9–8.7  2.5–4.8 
10  9  1.0  9.72  2  3.09  6.2 \(\times \) 10\(^{6}\)  2.5–39  7.9–123.8  7.9–8.7  2.5–4.8 
Figure 15 shows the path of the heavy particle (with \(\beta =0.3\)). The aspects mentioned in the preceding paragraph can more vigorously be seen in this figure. Since this particle is a heavy particle, it spends more time closer to the bottom, and thus experiences relatively stronger shear and turbulence. Hence, one would expect that heavy particles would be subjected to higher dispersion, just as in the case of steady flow.
Figure 21 suggests that the dependency of the dispersion coefficient on nondimensional amplitude gets weaker as the particles get heavier (i.e. for large \(\beta \) values). Likewise, for short amplitudes (a/ Open image in new window 40) the effect of \(\beta \) on the dispersion coefficient is more dominant than longer amplitudes.
In the study of Mazumder and Paul (2012), the bed condition defined for the particles was different than the present study, such that a settlement (bed absorbency) and reentrainment procedure was employed. Furthermore, their study does not clearly state an increase in the dispersion coefficient with increasing settling velocity. Thus, the two studies are not directly comparable. Nevertheless, the time variation of the variance found by Mazumder and Paul (2012) qualitatively agrees well with the present results (Figs. 17, 19).
Ng (2004) carried out an analytical study to find out the dispersion coefficients of neutrally buoyant particles in oscillatory flow. They used the free stream velocity and amplitude of the horizontal motion to nondimensionalise the dispersion coefficient. When converted to the terms of the present study, their findings for a/\(h=32\) and 96 come out to be \(\frac{D_1 }{\delta U_\mathrm{fm} }\approx \) 3.8 and 5, respectively. In the present model, those two values are found as 3.1 and 4.2. When compared, these findings are not radically diverse, given that the methodology followed by Ng (2004) and the present study is quite different.
5.4 Comparison with the oscillating tunnel experiments
As given in Sect. 4, three oscillating tunnel experiments were conducted (Table 1). The nondimensional settling velocity of the particles in these experiments was relatively high (namely \(\beta =3.6, 4.8\) and 7.2). Nevertheless, the bottom conditions were such that the particles did not deposit on the bottom when they came close to the bottom, and they were maintained in the flow all the time. These experiments were simulated in the numerical model and obtained results are compared with those of the experiments.
As is seen from the figures, the numerical simulations exhibit a general agreement with the experimental results for all the cases. For \(\beta =3.6\), the numerical model estimates the concentration at the upper regions slightly lower compared to the experimental results. As for \(\beta =7.2\), the estimate of the model is fairly more uniform while the experiment gave a more peaked result. It should, furthermore, be noted that this case also corresponds to the lowest wave Reynolds number (6 \(\times \) 10\(^{5})\), which is in the transitional regime. A probable explanation why the model seems to perform better for \(\beta =4.8\) is that, the number of particles captured in the experiments with \(\beta =4.8\) is almost two times larger than the other two experimental sets (Table 1). As the number of particles is increased, a better resolution of vertical concentration profile of particles could be obtained in the experiments, and thus a better comparison with the numerical results was maintained. These results suggest that the numerical model can capture the suspension behaviour of heavy particles reasonably well.
For all the three tested cases, the displacement of the particles in the oscillating tunnel experiments is approximately 30 % lower compared to the numerical simulation results. This is somehow expected, since the particles in the oscillating tunnel have a finite size and a higher specific mass than water, and thus they experience a certain lag each time the surrounding fluid is accelerated owing to their inertia.
It is seen that the dispersion behaviour in the experiments basically composed of an increasing trend of variance superimposed with some periodic variations, as depicted in Sect. 5.3. The time variation of the variance is not captured for large times due to the small sample size. Although the numerical model generally captures the constant increasing trend fairly well, the periodic variation amplitudes calculated by the model are significantly lower than those of the experiments, presumably attributed to the settling velocity of the particles being increased beyond the suspension threshold (i.e. \(\frac{w_\mathrm{s}}{U_\mathrm{fm} }>>1)\) .
Comparison of model results with experiments
Case  \(\frac{D_1 }{\delta \,U_\mathrm{fm} }\)  

\(Re_\mathrm{w}\)  a/h  \(\beta \)  Model  Experiments 
3.3 \(\times \) 10\(^{6}\)  16.43  3.6  5.1  4.9 
1.5 \(\times \) 10\(^{6}\)  10.67  4.8  6.0  7.2 
6.0 \(\times \) 10\(^{5}\)  6.72  7.2  6.1  9.6 
6 Longitudinal dispersion in wave boundary layer: practical application
In this section, a practical example will be given on how the presented numerical model can be used to simulate the longitudinal dispersion of heavy particles in wave boundary layers. The following case is considered:
Excavated material (fine sand with \(d_{50}=0.10\) mm) from a dredging operation is disposed to a coastal area at a depth of 10 m. The settling velocity of the material is \(w_\mathrm{s}=0.65\) cm/s. Some time after disposal, the material disperses under the action of storm waves with a rootmean square wave height of \(H_\mathrm{rms}=2.6\) m and zerocrossing period of \(T_\mathrm{z}=8\) s. This process will be simulated as follows by means of the presented numerical model.
First, the flow model is run for the determination of the mean flow and turbulence characteristics developing under the wave boundary layer. As the free stream velocity, the horizontal orbital velocity at the bed found from the linear wave theory, \(U_\mathrm{m}=1.01\) m/s, was used. The model upper boundary (i.e. frictionless lid) was set to \(h=1\) m. It can be shown by linear wave theory that the orbital velocity at the top of model upper boundary is practically equal to the one at the bed (\(U_{m (y=1m)} =1.02\) m/s). Thus, the oscillatory flow model results are directly applicable for this wave boundary layer case.
7 Conclusions

The results for steady flow case show that the model results are consistent with the existing information in the literature, and well represent the physics of the process for suspended particles unless the settling velocity is significantly high.

In the case of oscillatory flow, the variance of particle positions does not increase monotonously as in steady flow case, but it exhibits some periodic oscillations accompanied by a continuous trend of dispersion.

In the oscillatory flow case, the dispersion process is governed by the Rouse parameter as well as an additional nondimensional parameter representing the amplitude of the motion. Generally, in oscillatory flow the dispersion coefficients are significantly less than that of the steady flow. The primary reason for this decrease appears to be the lower shear and turbulence experienced by the particles. This is demonstrated graphically and numerically in the study.

For oscillatory flow, the dispersion coefficient increases with the settling velocity in a similar manner with the steady flow case. The reason for this increase is, again, that heavy particles moving closer to bed experience a higher shear (i.e. velocity gradient) and turbulence throughout their paths, which presumably disperses them farther and farther.

When the amplitude of the oscillatory motion increases, dispersion coefficient of neutrally buoyant particles increases significantly and asymptotically attains to the value of the steady, open channel, flow case. As the particles get heavier (i.e. as the Rouse parameter increases), the dispersion coefficient under oscillatory flow becomes less dependent on the amplitude of the motion.

The comparison of the present findings with the result of the experimental campaign, conducted in a Utunnel facility, showed a reasonable agreement although the settling velocity values in the experimental study were beyond the suspension regime.

Finally, the use of the present model was demonstrated with a practical case example in which the dispersion of fine sand under storm waves was studied.
Notes
Acknowledgments
This study has been partially funded by the European Commission, through the FP7 project Innovative MultiPurpose Offshore Platforms: Planning Design and Operation (MERMAID, G.A. No. 288710).
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