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Euler–Lagrange model for scour in front of vertical breakwater
Applied Ocean Research 34 (2012) 96–106
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Euler–Lagrange model for scour in front of vertical breakwater Fatemeh Hajivalie a , Abbas Yeganeh-Bakhtiary a,b,∗ , Hamid Houshanghi a , Hitoshi Gotoh c a
School of Civil Engineering, Iran University of Science and Technology (IUST), Tehran, Iran Hydro-environmental Research Centre, School of Engineering, Cardiff University, Cardiff, Wales, United Kingdom c Department of Civil and Earth Resources Engineering, Kyoto University, Kyoto, Japan b
a r t i c l e
i n f o
Article history: Received 11 December 2010 Received in revised form 24 August 2011 Accepted 25 September 2011 Available online 6 December 2011 Keywords: DEM model k–ε Turbulence closure model MBS model RANS equations Scour Standing waves Steady streaming
a b s t r a c t A two-dimensional Euler–Lagrange model was developed to study the scour in front of a vertical breakwater. The fluid phase was described via the Reynolds Averaged Navier–Stokes equations in conjunction with the k–ε turbulence closure model. The sediment phase was treated as an assembly of discrete sand grains and the scour was introduced as the motion of a granular media from the Lagrangian point of view. Motion of the sand grains was traced with a numerical code based on the so-called MBS model, in which the frequent interparticle collision described with a spring and dashpot system. Comparison between the numerical result and experimental measurement confirms that the numerical model successfully predicts the scour profile and steady streaming field. The results reveal that during the scour process sediment transport rate decrease and the scour/deposition pattern reaches to a semi-equilibrium shape. The sand grains were transported in three different modes; namely the hyper-concentrated flow, saltation and suspension mode. The concept of vertical momentum transfer is exploited to describe features of concurrently present of these different modes. It is evident that the vertical momentum transfers between upper recirculating cells of steady streaming moves the bed sediment into the upper layer; whereas the vertical motion of particles is not significantly active in the hyper-concentrated layer and the sediment grains momentum is preserved. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Various types of vertical breakwaters are constructed to dissipate the incident wave energy by reflecting almost all of the incident’s wave energy and provide a safe shelter area. The superposition of incident waves impinged on and the reflected waves from a vertical breakwater produce a series of standing waves, which hydrodynamics is rather complicated and consisted of the recirculating cells of steady streaming. This flow field of steady streaming interacts severely with the bed sediments resulting in local scour near vertical breakwater. The significance of steady streaming in scour near a vertical breakwater has provided the impetus for a number of investigations. For example, the generation of steady streaming and its effect on scour pattern in sand beds under the standing waves near a vertical breakwater is studied in details (e.g. [1–3]). The steady streaming is consisted of top and bottom recirculating cells as shown in Fig. 1 [21]. The scour pattern can be classified into two basic types as illustrated in Fig. 2. When the bed sediments is relatively fine, the sediments is mainly responding to the top recirculating cells and the scour pattern is identical to Fig. 2a; in contrast
∗ Corresponding author. Tel.: +98 21 77240096; fax: +98 21 77240398. E-mail address: [email protected] (A. Yeganeh-Bakhtiary). 0141-1187/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2011.09.006
the relatively coarse sediments are responding to the bottom cells and the scour pattern is similar to Fig. 2b [3]. In recent years there have been other studies on local scour at the trunk section of vertical breakwater in more complicated hydrodynamic condition. Gao and Inochi [4] experimentally studied local scour induced by broken waves in front of vertical breakwater. Lee and Mizutani [5] experimentally studied the scour at a vertical impermeable submerged breakwater in partial standing wave condition. However, the experimental study is rather challenging since it can be very expensive and time consuming and obtaining the robust data is rather difficult. In parallel the developments in computer hardware and numerical solution methods have persuaded the numerical models as a good alternative to study the scour process. Process-based numerical models directly simulate the major processes involved in scour using a hydrodynamic model coupled with a simple equation for the bed profile change. Chen [6] used a two-dimensional numerical wave flume based on an Arbitrary Lagrangian–Eulerian (ALE) description of the Navier–Stokes equations, to simulate the local scour in front of vertical breakwaters. The morphologic development of scoured bed was obtained by solving the equation of sediment continuity. Although the model was able to simulate the steady streaming cells with a good approximation, it was apparent that the model could just simulate the clockwise recirculating cells over boundary layer; the
F. Hajivalie et al. / Applied Ocean Research 34 (2012) 96–106
Fig. 1. Standing waves and steady streaming in front of vertical breakwater, Sumer and Fredsoe [22].
scour pattern also did not matched well with the experimental results. Gislason et al. [7] combined a 2D Navier–Stokes solver with a morphologic model to study scour and deposition in front of breakwaters. Although their results for scour in front of vertical breakwater were rather in consistent with the experimental data, the scour profile in front of the sloping breakwater did not follow the measured profiles. On the other hand, this process based models, which is categorized as the so-called single-phase models, were not able to provide much information on the scour mechanism as the result of flow-structure, flow-sediments and sediment–sediment interaction. The scour in front of a vertical breakwater is driven by relatively high-energy hydrodynamics; hence, rather than a single-phase phenomena, it is more realistic to consider the scour as consisting of the two-phase mixtures comprised of the bed sediment (sediment phase) and the flow hydrodynamics in front of vertical breakwater as the fluid phase. An
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important point that should be taken into account by the two-phase model is the momentum exchange between the two phases. In a two-phase flow model, the governing equations of fluid phase have been often described in Eulerian Form; however, the governing equations of the sediment phase may be expressed with the Eulerian or Lagrangian form. By coupling the governing equations of fluid and sediment phases, an Euler–Euler or Euler–Lagrange model can be obtained. Nevertheless the Euler–Lagrange two-phase model needs more computational time and memory; they are able to consider the probabilistic characteristics involving the motion of sediment grains, which is the most advantage of this kind of models. Extensive research on the Euler–Lagrange model has been carried out to study sediment transport (e.g. [8–14] to the best knowledge of the authors); however, the Euler–Lagrange twophase model has not yet been developed and applied for the studying of scour in front of a vertical breakwater. The main purpose of this paper was to develop and employ the Euler–Lagrange coupling two-phase flow modeling approach to investigate the scour process under standing waves formation and to describe the relevant hydrodynamic and sediment transport processes in front of vertical breakwater. The resulted steady streaming in front of a vertical breakwater, sediment transport modes during the scour process, momentum transfer between flow and sediment grains and vertical momentum transfer between sediment grains were discussed herewith in great detail. The model was developed based on Euler–Lagrange coupling of the governing equations for the sediment and fluid phases. The fluid phase was described via the Reynolds Averaged Navier–Stokes (RANS) equations with closuring a k–ε turbulence model to simulate the turbulence generation and dissipation processes. The free surface motion was simulated using the Volume Of Fluid (VOF) technique suggested by Hirt and Nichols [15]. Incident waves were generated by internal source functions inside the computational domain. To assess the numerical model performance with one-way coupling manner the two-phase model has been validated against the conventional formula and experiment for two different hydrodynamics conditions: (i) the sediment transport rate induced by unidirectional flow condition; and (ii) the standing wave-induced scour in front of vertical breakwater. The numerical results of the flow field as well as the scour profile showed a very good agreement with the Xie’s experimental data for standing wave-induced scour in front of vertical breakwater. The plan of this paper is as follows; after the introduction, in Section 2 the mathematical formulation of the fluid and sediment phase are described. Section 3 presents fluid phase and sediment phase numerical model. Model validation is presented in Section 4 and then numerical results for scour formation and sediment transport hydrodynamic under action of standing waves are discussed in the same section. Finally Section 5 concludes the main results of this study. 2. Mathematical formulation Mathematical formulations in the present model were founded based on the coupling of mass and momentum conservation equations of flow field with the sediment phase. The flow hydrodynamics and sediment transport were computed in a decoupled one-way manner. In the following the essential details of the model equations and their numerical solutions are described below. 2.1. Fluid phase
Fig. 2. Two types of scour pattern in front of a vertical breakwater Xie [3].
Yeganeh-Bakhtiary et al. [16] developed a two-dimensional model for the simulation of the hydrodynamics of standing wave
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Table 1 The k–ε model constants [17]. C
C1ε
C2ε
k
ε
0.09
1.44
1.92
1.0
1.3
formation in front of a vertical breakwater. The model is developed based on the Reynolds Averaged Navier–Stokes (RANS) equations, VOF technique, a k–ε turbulence model and internal source functions for generating of the incident waves enabling the investigation and analysis of steady streaming recirculating cell in front of a vertical breakwater. Here the previous model is extended to the two-phase flow model framework to simulate the local scour. The governing equations consisted of the continuity, momentum together with the equations for turbulent kinetic energy k and turbulence dissipation rate ε in the two-dimensional coordinates as follow: ∂Ui =0 ∂xi
(1)
∂Ui 1 ∂P ∂ ∂U −g+ + Ui i = − ∂xi ∂t ∂xj ∂xi +
∂ ∂xj
∂Uj ∂Ui + ∂xj ∂xi
∂
∂k ∂ + (kUi ) = ∂t ∂xi ∂xj ∂ε ∂ ∂ + (εUi ) = ∂t ∂xi ∂xj
pr = t 2
t = C
k2 , ε
∂Ui ∂xi
2
+
∂Ui ∂xj
= + t
∂Ui 2 ∂xi
s
+ CM
A3 dp3
dupi dt
=
[−fn cos ˛ij + fs sin ˛ij ]
+
CD A2 dp2 2
+
∂(U − upi ) CL A2 d2 |W − wpi | 2 ∂z
s
−
+ CM
s
(U − upi )2 + (W − wpi )2 (U − upi )
A3 dp3
dwpi
dt
− 1 gA3 dp3 +
× (W − wpi ) +
=
(8)
[−fn sin ˛ij + fs cos ˛ij ]
CD A2 dp2 2
(U − upi )2 + (W − wpi )2
∂(W − wpi ) CL A2 dp2 |U − Upi | 2 ∂z
t ε
∂k ∂xj
∂ε
2
∂xj
dp
d4 dωPi [fs ]j = 32 dt 2
+ pr − ε
(9)
ε ε2 + C1ε Pr − C2ε k k
(4)
(5)
(6)
(7)
2.2. Sediment phase The sediment phase was simulated using the two-dimensional form of the MBS-3D model of Yeganeh-Bakhtiary et al. [14], which is based on Cundall and Strack [18] idea for discrete regions. The present work extends the use of the MBS model to the framework of simulating the local scour for both suspension and non-suspension mode. The bed sediments were modeled as an assemble of spherical particles of the uniform diameter. A one-way coupling method is used to implement interactive forces between fluid flow and sediment particles. It is assumed that only the fluid phase exerts forces
(10)
j
(3)
here Ui is the mean velocity component of flow; i and j indexes represent horizontal and vertical direction, respectively; P is the mean pressure; g is the acceleration of gravity; is the fluid density; is the effective viscosity; and t are respectively kinematic and kinetic eddy viscosity; k is the turbulence kinetic energy; pr is the production of turbulence kinetic energy; and ε is the turbulence dissipation rate. The model constants are set according to Launder and Spalding [17] presented in Table 1 (for more details please refer to Yeganeh-Bakhtiary et al. [16]). The free surface motion is tracked by the VOF technique of Hirt and Nichols [15], which satisfies both the kinematic and dynamic free surface conditions. The conservation of F or the volume fraction of fluid is expressed as follows: ∂F ∂F =0 + Ui ∂xi ∂t
(2)
t + k
+
on the sediment phase, while the fluid flow remains uninfluenced. Such an assumption is accurate because: (i) as shown experimentally by Govan et al. (1989) and Yeganeh-Bakhtiary (1997) the presence of moving particles does not significantly modify the fluid flow at its upper part; (ii) within the hyper concentrated sediment layer, the velocity difference between the fluid flow and sediment particles is negligible. Therefore, the equations of motion of the ith particle on the vertically 2D coordinates are as follow:
24 CD = CD∞ + , Rep
CD∞ = 0.4
(11)
where s is the sediment density; A2 , A3 are the two and three dimensional geometrical coefficient of sediment; CM is the added mass coefficient (= 0.5); CD is the drag coefficient; CL is the lift coefficient (=4/3) [19]; dp is the sediment particle diameter; upi , wpi are the streamwise and vertical component of particle velocity; fn , fs are the normal and tangential components of the force acting between ith and jth particles; ωpi is the rotational angle of the ith particle; and Rep = (up dp /) is the sediment grain Reynolds number. In case of uniform particles the assessment of contacting particle is formulated by:
(xi − xj )2 + (yi − yj )2 ≤ dp
(12)
here (xi , xj ), (zi , zj ) are the coordinate of centroid of the ith and jth particles, dp is the particle diameter; and is the constant around 1.0. The acting force in normal and tangential direction between two contacting particles ith and jth can be estimated by the following relations: fn (t) = en (t) + dn (t)
(13)
fs (t) = es (t) + ds (t)
(14)
en (t) = min[en (t − t) + kn n , en max ]
(15)
es (t) = min[es (t − t) + ks s , es max ]
(16)
dn (t) = −n · n
(17)
ds (t) = −s · s
(18)
s = − n −
d t(ωpi + ωpj ) × n 2
(19)
where en , es are the forces working on the spring; dn , ds are the damping forces acting on the dashpot; n , s are the displacement of particles during the time step of the calculation t; kn , ks are the spring constants; n , s are the damping coefficient. In this model sediment grains were assumed to be non-cohesive
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Turbulent intensities are estimated according to Nezu [21] as follow:
u¯ 2 = 1.108 ×
¯ 2 = 0.336 × w
k
(27)
k
(28)
¯ are averaged turbulence velocities respectively in here u¯ and w horizontal and vertical direction. 3.2. Sediment phase domain Fig. 3. Numerical model configuration of Xie’s experiment [3].
grains, therefore the joint between contacting particles did not resist against the tension forces. On the other hand, a shear stress limit was utilized in the local tangential direction, the joint slips by exceeding this limit. The joint characteristics can be defined as: fn (t) = fs (t) = 0 when nen (t) < 0
(20)
|fs (t)| = |en (t)| when |es (t)| > |es (t)|
(21)
here is the coefficient of friction (= 0.55). The time step t should be proportional to critical time step or tc to satisfy DEM model stability. For a single mass-spring system with a single degree of freedom based on the mass m, t and tc can be estimated as:
t = tc /20;
tc = 2
m m ⇒ kn = 2 2kn
2 2 tc
(22)
where m is the mass of particle. In this model, the time step is adopted as t = 0.0002 s; thus kn and ks were estimated by Eq. (22) and the theory of elasticity as below: ks =
kn 2(1 + )
(23)
here (= 0.3) is the Poisson’s ratio. The damping coefficients were estimated from the critical damping conditions:
n = ˛cn 2
mkn ;
s =
n 2(1 + )
(24)
here ˛cn (= 1.0) was a calibrating coefficient. For more information please refer to Yeganeh-Bakhtiary et al. [14]. 3. Numerical method 3.1. Fluid phase
In addition to the fluid field, the sediment phase domain has to be specified. Before the main calculations, the packing calculation was executed to determine the initial position of particles. During this process, velocity of falling particles was monitored to assess the convergence of packing calculation. Fig. 4 shows a schematically sketch of the sediment domain placed in flow domain in front of a vertical breakwater. For defining the sediment motion, in fact two different domains are set together: an Eulerian domain for the fluid phase and a Lagrangian one for the sediments. The fluid domain is penetrated to the sediment domain. Thus, the fluid phase at its bottom grids is assumed to be containing with the sediment particles. The sediment transport occurs due to the interaction forces between the carrier fluid and sediment bed. In other words, the sediment particles is able to move freely in the Eulerian grids and the surrounding fluid velocity for each particle can be computed based on the particle location in the Eulerian grids. In the sediment phase the packing is essential to determine the initial position of sediment particles before starting calculations; thus two different sets of particles in the packing procedure was defined, namely main sediment particles and solid wall particles. The main particles were initially arranged with leaving a 0.001dp gap between each other. Prior to the main calculation phase, a preliminary packing procedure was executed to determine the locations of sediment particles. In other word, the sediment particles were subjected to gravity force in packing process to fill initial porosity leading to a more natural arrangement of bed particles. During the packing process, the velocity of moving particles was monitored to assess the convergence of packing calculation. At wall boundaries, mechanical characteristics of particles were set twice as the main particles in the domain; since these particles were set to have no displacement during interparticle contacts, therefore static contact characteristics were replaced with kinetic ones. knwall = 2kn ;
Fig. 3 gives a sketch of the fluid-phase domain for the configuration of Xie [3] experiment. The initial and boundary condition is quite similar to the last model of Yeganeh-Bakhtiary et al. [16] for the standing wave formation in front of a vertical breakwater. To be more concise herewith only the numerical wave generation in the flow domain is described. At the inlet boundary, the weakly reflective boundary condition, recommended by Petit et al. [20] was used for generating the incident waves and simultaneously absorbing the reflected waves from the vertical breakwater to prohibit the collision (intermixing) of the generated and reflected waves that propagate in the direction opposite the zone of interest, as follow: ∂Rr ∂Rr + Cr =0 ∂t ∂x
(25)
Rr = Rt − Rin
(26)
here Cr is the celerity of reflected wave; Rr , Rt and Rin are the variable associated with the reflected, computed and the theoretical wave values represents velocity, pressure and free surface displacement and Rin was calculated by the Stokes second order theory.
kswall = 2ks ;
wall = 2
wall = 2n ; n
wall = 2s ; s (29)
In every 0.1 s, the fluid velocities were fed to the sediment phase model, during each fluid phase time increment, the fluid velocity was considered to be constant and sediment particles were moving under action of a semi-steady flow. For every single particle, the flow velocities were calculated by locating the particle in the flow model mesh cells, then the orbital and the turbulence velocities could be estimated by interpolation and the lift and drag forces were calculated then. 4. Results and discussion 4.1. Model validation To assess the numerical model accuracy with one-way coupling approach, the model should be validated in the simple case studies. As mentioned before, the developed two-phase model consists of two parts; namely fluid flow model and sediment phase
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Fig. 4. Sediment domain placed in flow domain.
Müller experimental formula [23] to estimate the sediment transport described below: qb
q∗b =
∗ =
u2∗ ((s /) − 1)gd
((s /) − 1)gd3
∗ n = a( ∗ − cr )
(30)
(31)
here qb is the sediment transport rate, a and n are the empirical ∗ is the criticonstants; * is the Shields parameter; and cr cal Shields number. Based on the modified Shields diagram
Fig. 5. Computational domain for simulation of sediment transport under action of unidirectional flow.
model. The fluid flow model was extensively validated by YeganehBakhtiary et al. [16] for the simulation of fully and partially standing wave hydrodynamics in front of a vertical breakwater in terms of the mean and turbulent flow characteristics. Here the two-phase model was validated against two different hydrodynamics conditions, namely the sediment transport rate induced by unidirectional flow condition, and the standing wave-induced scour in front of vertical breakwater. For the former case, the sediment transport rate under the steady unidirectional flow was simulated and the results were compared with Meyer-Peter and Müller [23] experimental formula. The latter case, the wave-induced scour in front of vertical breakwater was simulated and were compared with experimental data of Xie [3]. In both cases the comparisons show a good agreement between the numerical results and experimental formula/data. Fig. 5 shows the numerical computational domain for simulation of sediment transport under action of steady unidirectional flow. In this simulation the sediment domain consist of uniform spheres, which were subjected to three different unidirectional flow with three different Shields parameter equal to 0.2, 0.4 and 0.8. Table 2 summarizes the physical characteristics of these three test cases. The results then were compared with Meyer-Peter and
Fig. 6. Sediment transport rate versus Shields parameter.
Table 2 Physical characteristics for simulation of transport under unidirectional flow. Test no.
s (kg/m3 )
D (mm)
U (m/s)
*
u*
1 2 3
2650 2650 2650
0.12 0.12 0.12
0.3 0.3 0.3
0.2 0.4 0.8
0.019 0.027 0.038
Fig. 7. Steady streaming pattern induced by standing waves in front of vertical breakwater.
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Table 3 Physical characteristics of Xie [3] experiment. H (m)
T (s)
h (m)
L (m)
H/gT2
d/T2
D50 (mm)
Sand type
Smax (cm)
0.05
1.17
0.3
1.71
0.0017
0.010
0.106
Relatively fine
1.3
recommended by Madsen and Grant [24], the critical Shields parameter is set to cr = 0.055, and the conventional values of a = 8.5 and n = 1.5 were used for the Meyer-Peter and Müller formula [23]. However, the modified constants as a = 7.5 and n = 1.1
were also adapted to the numerical results at higher values of shear stress (* > 0.4). Fig. 6 confirms good agreement between numerical results from present model and the bed-load transport formula.
Fig. 8. Development of scour/deposition in front of vertical breakwater under action of standing waves.
Fig. 9. (A) Wave induced scour pattern in front of vertical breakwater by Xie [3] and (B) comparison between model and experimental result by Xie [3] (diamonds).
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Fig. 10. Particle movement during a wave period in the zone between L/8 and L/4.
For the second case, the numerical two-phase model has been compared with the experimental results reported by Xie [3], which has been used as a benchmark case for evaluating scour computation induced by standing wave in front of vertical breakwater. Xie’s experiment was conducted in a 38 m long, 0.8 m wide and 0.6 m deep wave flume. At the beginning of the flume, the water depth was equal to 0.45 m; it then reduced to 0.3 m at the flat bed near the vertical breakwater with a 1:30 slope. Wave height and period were, respectively, 0.05 m and 1.17 s. The sediment mean diameter (dp50 ) was 0.106 mm and s = 2650 kg/m3 ; in this test the sand grains were relatively fine and suspension mode was
observed. In the numerical simulation, to reduce the computation time, the domain length was shortened to 14.5 m, which is discrete by 1 cm × 2 cm rectangular meshes. For the same reason, particles diameter was adjusted to 5.0 mm. To avoid any conflict between the experiment and the numerical simulation, Shields parameter in the numerical simulation was equal to the experiment. Table 3 presents the numerical configuration and summaries the physical characteristics of the Xie [3] experiment. Numerical simulation of scour relies on accurate simulation of flow pattern. Hence, the flow model was initially investigated to examine its capability in the prediction of flow pattern in front of a
F. Hajivalie et al. / Applied Ocean Research 34 (2012) 96–106
vertical breakwater. The fluid model have been validated with the experimental data of Xie [3] (see [16]) for standing waves orbital velocity. Generally, there was good agreement between the model results and different experimental data sets. Fig. 7 plots the steady streaming pattern generated by standing waves in front of the vertical breakwater. The orbital velocity of water particles was averaged during a whole period after the standing waves developed. As can be seen, the figure clearly shows a regular system of the recirculating cells of steady streaming generated in front of the vertical wall. The recirculating cells consist of top and bottom cells; while the bottom cells were attributed to the formation of bottom boundary layer as already illustrated in Fig. 1. It has been conjectured that the local scour is directly tied up to the dynamics of recirculating cells system of the steady streaming formation in front of the vertical breakwater [22]. To provide a better understanding of local scour formation, snapshots of scour/deposition development in the area extended from the near side to the distance of L/2 from the breakwater is depicted in Fig. 8. Rather long time lapse among snapshots was chosen to show how the bed sediments kept their motion. As shown
Fig. 11. Different sediment transport modes.
Fig. 12. Orbital and averaged sediment particle velocity during a wave period.
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Fig. 13. Averaged particle velocity in zero flow velocity zone during a wave period.
in the figure, the bed sediments responded to the top recirculating cells and after few wave periods, two sand bars were formed. The former sand bar formed at the vicinity of breakwater and the latter one at the distance of L/2 from the breakwater; consequently, the scour hole shaped between two bars at about L/4 from the breakwater. By continuing the influence of standing waves, the scour hole was getting deepen and after about 12T, it nearly reached the maximum scour depth. Then the bed sediments movement was decreasing with time; hence, after about 24T the maximum scour depth was almost achieved. It is apparent that after 26T a semi-equilibrium condition was reached in which the bed sediment movement was not changing the scour/deposition pattern of bed and the maximum scour depth could be estimated accordingly. In Fig. 9 the numerical result on the scour profile was compared with that of the Xie [3] experimental data. As can be seen, the scour pattern estimated by the model agrees very well at both the close vicinity and the distance of L/4 from the breakwater, a slight deviation, however, from the experimental data could be observed at the distance of about L/2 away from the breakwater. In other words, the simulation result fits very well with the scour pattern obtained
from the experiment. The maximum scour depth in the selected test case was 1.3 cm; while, the calculated maximum scour depth was 1.35 cm agrees well with the experiment. Qualitative comparison between Figs. 2a and 9 reveals that the scour pattern is identical to relatively fine sand pattern one. Thus the model was successful in the calculation of bedform and maximum scour depth. One of the main advantages of a two-phase flow model is its capability in estimating the scour pattern, as well as the sediment transport hydrodynamic in details. 4.2. Sediment transport hydrodynamic The hydrodynamic of sediment transport induced by standing waves in front of vertical breakwater is discussed in this section, Fig. 10 shows the snapshots of particle movements during a wave period with the time interval of T/6 in a distance between L/8 and L/4 away from the breakwater, in which the most particle movement was observed in this zone. The fluid velocity and the horizontal velocity of moving sediment are presented by the vectors and different colors, respectively. As it is observed in the figure,
F. Hajivalie et al. / Applied Ocean Research 34 (2012) 96–106
the orbital velocity started to increase from zero to its maximum at the lee ward direction of vertical breakwater, but the bed sediments shows a slight time lag to start their motion and to accelerate. The particle velocity and number of moving particles start to increase simultaneously with time. At the beginning the bed sediments keep their motion concurrently in the hyper concentrated layer and saltating particles over it. With increasing the wave orbital velocity, some bed sediment take parted from the bed and started to suspend in the wave flow domain after one quarter of wave period. In the mid period, the wave orbital velocity changes its direction to the seaward then it took a while for the moving sediment to accelerate in reverse direction. A possible illustration of the various different sediment transport modes is given in Fig. 11. In lower regions of snapshots, where the particles mainly move in hyper concentrated layer, the presence of hyper-concentrated layers of particles prevents the active sediment motion. Thus, a rather slow pattern of motion is formed in this region and the vertical displacements of the sand grains are negligible. In upper regions of snapshots, in which the saltating particles ride over the protuberance of lower particles, the concentration of moving particles is relatively low. Therefore, the motion of individual particle in this region is less frequently affected by its neighboring particles, compared to what takes place in the hyper-concentrated layer. The topmost region contains the least particle concentration where the suspended particles move more freely than the saltating particles and rarely collide to each other. These particles have the highest velocity and respond to the top recirculating cells very actively. The predominant parameter affecting mode of sediment transport is the momentum transfer due to fluid–sediment interactions between phases and collision stresses in the sediment phase. In the hyper-concentrated layer the vertical motion of sediments is negligible since the high sediment concentration induces very frequent interparticle collisions; therefore, the interparticle stresses in this layer are preserved. By contrast, in the upper layers, the concentration is low and vertical velocity of sediments Therefore, in these regions, according to Gotoh et al. [10] and Yeganeh-Bakhtiary et al. [11], there will be two paths of vertical momentum transfer due to: (i) vertical mixing with fluid; and (ii) vertical motion of grains keeping interaction with surrounding fluid. Since the vertical motions of saltating and suspended grains are significantly active, the exchanged momentum is well-mixed between the upper and lower sediment layers. In other words, the interphase momentum transfer is promoted much because frequent interparticle collisions are less likely to take place; hence, the fluid–particle interaction is the predominant mechanism of transport at the upper regions. Fig. 12 illustrates the relative averaged horizontal velocity of sediments and the relative horizontal orbital velocity, U/Ufmax , versus the relative height, Z/dp , in the distance of L/8 away from the breakwater. In this figure, the height the zone between 8dp to 12dp represents the hyper-concentrated zone; between 12dp to 14dp is the saltation zone and upper than 14dp is the suspension zone. The panels show that by starting a wave period, it took a small time less than 0.2 s (≈ T/6) for sediments to accelerate and follow the flow movement. Nevertheless, when the flow direction changes the same situation was observed, for example in T/2 sediment velocity in the suspension zone was even against the flow velocity. The graphs also reveal that sediment velocity can be half of flow velocity in maximum. The interesting thing about these graphs is the region of zero flow velocity. Fig. 13 shows the zero flow velocity zones individually for each graph as illustrated in Fig. 12. This zero flow zone represent bed region which flow velocity does not interfere inside it; however, it is readily seen in the all graphs, the sediment particles have a small velocity. This velocity is not produced by fluid flow, but by the vertical momentum transform from the upper moving particles.
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Fig. 14. Suspension load ratio in one wave period.
Fig. 14 depicts the percentage of the suspension load to the total load of sediment transport and the horizontal orbital velocity in the distance of L/8 from the breakwater during a wave period, the orbital velocity was measured 5 cm over the bed. The graph seems to have two maximum points, one after the T/4 when the flow velocity is lee ward and the other before 3T/4 when the velocity is see ward. The maximum percentage of suspended load is about 20%. 5. Conclusion A two-phase flow model has been developed to analyze numerically standing waves induced scour in front of a vertical breakwater. This model was formulated for the two key physical processes involved: the recirculating steady streaming formation and the sediment transport. The flow model is based on the mass and momentum conservation equations, the VOF technique for tracking the free surface motion and a k–ε turbulence closure model for describing the generation and dissipation processes of turbulence. The sediment phase was extended based on the 2D version of MBS-3D model of Yeganeh-Bakhtiary et al. [14]. In summary, the simulation results agree well with the experimental study of Xie [3]. The model was successful to calculate maximum scour depth as well as bedform profile after the semi-equilibrium profile was obtained. Moreover, in selected experiment for the simulation, sediments were relatively fine and therefore sediment suspension should occur during the scour, the numerical simulation results showed that the dominant sediment transport mode is hyperconcentrated with suspension which confirms the reliability of the model in simulation of scour mechanism. According to the analysis of this study, the following conclusions are drawn: • Most of the numerical models available for simulating scour in front of vertical breakwaters are based on single-phase flow
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models. An Euler–Lagrange two-phase flow model provides a powerful simulation tool for studying the sediment transport and hydrodynamics in the scour process. • Sediment transport rate is unsteady and non-uniform during a wave period and scour process; however as time elapsing the bed sediment movement decrease and a semi-equilibrium stage of scour is reachable for scour in front of a vertical breakwater. • There is a time lag less than 0.2 s (≈ T/6) between flow velocity and sediment velocity mainly due to the sediment inertia. Maximum suspension rate occurs with a time lag after T/4 and 3T/4 of each wave period where the maximum orbital velocity occurs. The maximum suspension ratio is about 20%. • Particles in hyper-concentrated layer – where the flow velocity is negligible and the flow does not interfere – move under action of upper moving particles that vertically transform momentum to the lower particles. References [1] Carter TG, Liu LFP, Mei CC. Mass transport by waves and offshore sand bedforms. Journal of Waterway Harbors and Coastal Engineering 1973;99(WW2):165–84. [2] Mei CC. The applied dynamic of ocean surface waves. World Scientific; 1989, 740 pp. [3] Xie SL. Scouring pattern in front of vertical breakwaters and their influence on the stability of the foundation of the breakwaters. Delft: Delft University of Technology; 1981. p. 61. [4] Gao X, Inouchi K. The characteristics of scouring and deposition in front of vertical breakwaters by broken clapotis. Coastal Engineering Journal 1998;40.(1). [5] Lee KH, Mizutani N. Experimental study on scour occurring at a vertical impermeable submerged breakwater. Applied Ocean Research 2008;30(2):92–9. [6] Chen B. The numerical simulation of local scour in front of a vertical-wall breakwater. In: Proceedings of the conference of global Chinese scholars on hydrodynamics. 2006. p. 134–8. [7] Gislason K, Fredsoe J, Sumer BM. Flow under standing waves Part 2. Scour and deposition in front of breakwaters. Coastal Engineering 2009;56(3):363–70. [8] Wiberg PL, Smith JD. Model for calculating bed-load transport of sediment. Journal of Hydraulic Engineering, ASCE 1989;115(1):101–23.
[9] Jiang Z, Haff PK. Multiparticle simualtion methods applied to the micromechanism of bed-load transport. Water Resources Research 1993;29(2):399–412. [10] Gotoh H, Tsujimoto T, Nakagawa H. Numerical model of interphase momentum transfer and interparticle collision in bed-load layer. APD-IAHR; 1994. pp. 565–72. [11] Yegane-Bakhtiary A, Gotoh H, Sakai T. Applicability of Euler–Lagrange coupling multiphase-flow model to bed-load transport under high bottom shear. Journal of Hydraulic Research 2000;38(5):389–98. [12] Drake TG, Calantoni J. Discrete particle model for sheet flow sediment transport in the nearshore. Journal of Geophysical Research-Oceans 2001;106(C9):19859–68. [13] Calantoni J, Holland KT, Drake TG. Modelling sheet-flow sediment transport in wave-bottom boundary layers using discrete-element modelling. Philosophical Transactions of the Royal Society of London Series A – Mathematical Physical and Engineering Sciences 2004;362(1822):1987–2001. [14] Yeganeh-Bakhtiary A, Shabani B, Gotoh H, Wang SSY. A three-dimensional distinct element model for bed-load transport. Journal of Hydraulic Research 2009;47(2):203–12. [15] Hirt C, Nichols BD. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 1981:39. [16] Yeganeh-Bakhtiary A, Hajivalie F, Hashemi-Javan A. Steady streaming and flow turbulence in front of vertical breakwater with wave overtopping. Applied Ocean Research 2010;32(1):91–102. [17] Launder BE, Spalding DB. The numerical computation of turbulent flow. Journal of Computational Mechanic in Application of Mechanic and Engineering 1971;3. [18] Cundall PA, Strack OD. A discrete numerical model for granular assemblies. Géotechnique 1979;29(1):47–65. [19] Dong P, Zhang KF. Two-phase flow modelling of sediment motions in oscillatory sheet flow. Coastal Engineering 1999;36(2):87–109. [20] Petit HAH, van Gent MRA, van den Boscj P. Numerical simulation and validation of plunging breakers using a 2D Navier–Stokes model. In: 24th international conference on coastal engineering. 1994. p. 511–24. [21] Nezu I. Turbulent structure in open-channel flows. Kyoto, Japan: Kyoto University; 1977. [22] Sumer BM, Fredsoe J. Experimental study of 2D scour and its protection at a rubble-mound breakwater. Coastal Engineering 2000;40(1):59–87. [23] Meyer-Peter E, Müller R. Formulas for bed-load transport. In: Proceedings of the 2nd IAHR congress: 39–64. 1948. [24] Madsen OS, Grant WD. Quantitative description of sediment transport by waves. In: Proceedings of the 15th ASCE coastal engineering conference: 1093–1112. 1976.
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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Euler–Lagrange model for scour in front of vertical breakwater Fatemeh Hajivalie a , Abbas Yeganeh-Bakhtiary a,b,∗ , Hamid Houshanghi a , Hitoshi Gotoh c a
School of Civil Engineering, Iran University of Science and Technology (IUST), Tehran, Iran Hydro-environmental Research Centre, School of Engineering, Cardiff University, Cardiff, Wales, United Kingdom c Department of Civil and Earth Resources Engineering, Kyoto University, Kyoto, Japan b
a r t i c l e
i n f o
Article history: Received 11 December 2010 Received in revised form 24 August 2011 Accepted 25 September 2011 Available online 6 December 2011 Keywords: DEM model k–ε Turbulence closure model MBS model RANS equations Scour Standing waves Steady streaming
a b s t r a c t A two-dimensional Euler–Lagrange model was developed to study the scour in front of a vertical breakwater. The fluid phase was described via the Reynolds Averaged Navier–Stokes equations in conjunction with the k–ε turbulence closure model. The sediment phase was treated as an assembly of discrete sand grains and the scour was introduced as the motion of a granular media from the Lagrangian point of view. Motion of the sand grains was traced with a numerical code based on the so-called MBS model, in which the frequent interparticle collision described with a spring and dashpot system. Comparison between the numerical result and experimental measurement confirms that the numerical model successfully predicts the scour profile and steady streaming field. The results reveal that during the scour process sediment transport rate decrease and the scour/deposition pattern reaches to a semi-equilibrium shape. The sand grains were transported in three different modes; namely the hyper-concentrated flow, saltation and suspension mode. The concept of vertical momentum transfer is exploited to describe features of concurrently present of these different modes. It is evident that the vertical momentum transfers between upper recirculating cells of steady streaming moves the bed sediment into the upper layer; whereas the vertical motion of particles is not significantly active in the hyper-concentrated layer and the sediment grains momentum is preserved. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Various types of vertical breakwaters are constructed to dissipate the incident wave energy by reflecting almost all of the incident’s wave energy and provide a safe shelter area. The superposition of incident waves impinged on and the reflected waves from a vertical breakwater produce a series of standing waves, which hydrodynamics is rather complicated and consisted of the recirculating cells of steady streaming. This flow field of steady streaming interacts severely with the bed sediments resulting in local scour near vertical breakwater. The significance of steady streaming in scour near a vertical breakwater has provided the impetus for a number of investigations. For example, the generation of steady streaming and its effect on scour pattern in sand beds under the standing waves near a vertical breakwater is studied in details (e.g. [1–3]). The steady streaming is consisted of top and bottom recirculating cells as shown in Fig. 1 [21]. The scour pattern can be classified into two basic types as illustrated in Fig. 2. When the bed sediments is relatively fine, the sediments is mainly responding to the top recirculating cells and the scour pattern is identical to Fig. 2a; in contrast
∗ Corresponding author. Tel.: +98 21 77240096; fax: +98 21 77240398. E-mail address: [email protected] (A. Yeganeh-Bakhtiary). 0141-1187/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2011.09.006
the relatively coarse sediments are responding to the bottom cells and the scour pattern is similar to Fig. 2b [3]. In recent years there have been other studies on local scour at the trunk section of vertical breakwater in more complicated hydrodynamic condition. Gao and Inochi [4] experimentally studied local scour induced by broken waves in front of vertical breakwater. Lee and Mizutani [5] experimentally studied the scour at a vertical impermeable submerged breakwater in partial standing wave condition. However, the experimental study is rather challenging since it can be very expensive and time consuming and obtaining the robust data is rather difficult. In parallel the developments in computer hardware and numerical solution methods have persuaded the numerical models as a good alternative to study the scour process. Process-based numerical models directly simulate the major processes involved in scour using a hydrodynamic model coupled with a simple equation for the bed profile change. Chen [6] used a two-dimensional numerical wave flume based on an Arbitrary Lagrangian–Eulerian (ALE) description of the Navier–Stokes equations, to simulate the local scour in front of vertical breakwaters. The morphologic development of scoured bed was obtained by solving the equation of sediment continuity. Although the model was able to simulate the steady streaming cells with a good approximation, it was apparent that the model could just simulate the clockwise recirculating cells over boundary layer; the
F. Hajivalie et al. / Applied Ocean Research 34 (2012) 96–106
Fig. 1. Standing waves and steady streaming in front of vertical breakwater, Sumer and Fredsoe [22].
scour pattern also did not matched well with the experimental results. Gislason et al. [7] combined a 2D Navier–Stokes solver with a morphologic model to study scour and deposition in front of breakwaters. Although their results for scour in front of vertical breakwater were rather in consistent with the experimental data, the scour profile in front of the sloping breakwater did not follow the measured profiles. On the other hand, this process based models, which is categorized as the so-called single-phase models, were not able to provide much information on the scour mechanism as the result of flow-structure, flow-sediments and sediment–sediment interaction. The scour in front of a vertical breakwater is driven by relatively high-energy hydrodynamics; hence, rather than a single-phase phenomena, it is more realistic to consider the scour as consisting of the two-phase mixtures comprised of the bed sediment (sediment phase) and the flow hydrodynamics in front of vertical breakwater as the fluid phase. An
97
important point that should be taken into account by the two-phase model is the momentum exchange between the two phases. In a two-phase flow model, the governing equations of fluid phase have been often described in Eulerian Form; however, the governing equations of the sediment phase may be expressed with the Eulerian or Lagrangian form. By coupling the governing equations of fluid and sediment phases, an Euler–Euler or Euler–Lagrange model can be obtained. Nevertheless the Euler–Lagrange two-phase model needs more computational time and memory; they are able to consider the probabilistic characteristics involving the motion of sediment grains, which is the most advantage of this kind of models. Extensive research on the Euler–Lagrange model has been carried out to study sediment transport (e.g. [8–14] to the best knowledge of the authors); however, the Euler–Lagrange twophase model has not yet been developed and applied for the studying of scour in front of a vertical breakwater. The main purpose of this paper was to develop and employ the Euler–Lagrange coupling two-phase flow modeling approach to investigate the scour process under standing waves formation and to describe the relevant hydrodynamic and sediment transport processes in front of vertical breakwater. The resulted steady streaming in front of a vertical breakwater, sediment transport modes during the scour process, momentum transfer between flow and sediment grains and vertical momentum transfer between sediment grains were discussed herewith in great detail. The model was developed based on Euler–Lagrange coupling of the governing equations for the sediment and fluid phases. The fluid phase was described via the Reynolds Averaged Navier–Stokes (RANS) equations with closuring a k–ε turbulence model to simulate the turbulence generation and dissipation processes. The free surface motion was simulated using the Volume Of Fluid (VOF) technique suggested by Hirt and Nichols [15]. Incident waves were generated by internal source functions inside the computational domain. To assess the numerical model performance with one-way coupling manner the two-phase model has been validated against the conventional formula and experiment for two different hydrodynamics conditions: (i) the sediment transport rate induced by unidirectional flow condition; and (ii) the standing wave-induced scour in front of vertical breakwater. The numerical results of the flow field as well as the scour profile showed a very good agreement with the Xie’s experimental data for standing wave-induced scour in front of vertical breakwater. The plan of this paper is as follows; after the introduction, in Section 2 the mathematical formulation of the fluid and sediment phase are described. Section 3 presents fluid phase and sediment phase numerical model. Model validation is presented in Section 4 and then numerical results for scour formation and sediment transport hydrodynamic under action of standing waves are discussed in the same section. Finally Section 5 concludes the main results of this study. 2. Mathematical formulation Mathematical formulations in the present model were founded based on the coupling of mass and momentum conservation equations of flow field with the sediment phase. The flow hydrodynamics and sediment transport were computed in a decoupled one-way manner. In the following the essential details of the model equations and their numerical solutions are described below. 2.1. Fluid phase
Fig. 2. Two types of scour pattern in front of a vertical breakwater Xie [3].
Yeganeh-Bakhtiary et al. [16] developed a two-dimensional model for the simulation of the hydrodynamics of standing wave
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Table 1 The k–ε model constants [17]. C
C1ε
C2ε
k
ε
0.09
1.44
1.92
1.0
1.3
formation in front of a vertical breakwater. The model is developed based on the Reynolds Averaged Navier–Stokes (RANS) equations, VOF technique, a k–ε turbulence model and internal source functions for generating of the incident waves enabling the investigation and analysis of steady streaming recirculating cell in front of a vertical breakwater. Here the previous model is extended to the two-phase flow model framework to simulate the local scour. The governing equations consisted of the continuity, momentum together with the equations for turbulent kinetic energy k and turbulence dissipation rate ε in the two-dimensional coordinates as follow: ∂Ui =0 ∂xi
(1)
∂Ui 1 ∂P ∂ ∂U −g+ + Ui i = − ∂xi ∂t ∂xj ∂xi +
∂ ∂xj
∂Uj ∂Ui + ∂xj ∂xi
∂
∂k ∂ + (kUi ) = ∂t ∂xi ∂xj ∂ε ∂ ∂ + (εUi ) = ∂t ∂xi ∂xj
pr = t 2
t = C
k2 , ε
∂Ui ∂xi
2
+
∂Ui ∂xj
= + t
∂Ui 2 ∂xi
s
+ CM
A3 dp3
dupi dt
=
[−fn cos ˛ij + fs sin ˛ij ]
+
CD A2 dp2 2
+
∂(U − upi ) CL A2 d2 |W − wpi | 2 ∂z
s
−
+ CM
s
(U − upi )2 + (W − wpi )2 (U − upi )
A3 dp3
dwpi
dt
− 1 gA3 dp3 +
× (W − wpi ) +
=
(8)
[−fn sin ˛ij + fs cos ˛ij ]
CD A2 dp2 2
(U − upi )2 + (W − wpi )2
∂(W − wpi ) CL A2 dp2 |U − Upi | 2 ∂z
t ε
∂k ∂xj
∂ε
2
∂xj
dp
d4 dωPi [fs ]j = 32 dt 2
+ pr − ε
(9)
ε ε2 + C1ε Pr − C2ε k k
(4)
(5)
(6)
(7)
2.2. Sediment phase The sediment phase was simulated using the two-dimensional form of the MBS-3D model of Yeganeh-Bakhtiary et al. [14], which is based on Cundall and Strack [18] idea for discrete regions. The present work extends the use of the MBS model to the framework of simulating the local scour for both suspension and non-suspension mode. The bed sediments were modeled as an assemble of spherical particles of the uniform diameter. A one-way coupling method is used to implement interactive forces between fluid flow and sediment particles. It is assumed that only the fluid phase exerts forces
(10)
j
(3)
here Ui is the mean velocity component of flow; i and j indexes represent horizontal and vertical direction, respectively; P is the mean pressure; g is the acceleration of gravity; is the fluid density; is the effective viscosity; and t are respectively kinematic and kinetic eddy viscosity; k is the turbulence kinetic energy; pr is the production of turbulence kinetic energy; and ε is the turbulence dissipation rate. The model constants are set according to Launder and Spalding [17] presented in Table 1 (for more details please refer to Yeganeh-Bakhtiary et al. [16]). The free surface motion is tracked by the VOF technique of Hirt and Nichols [15], which satisfies both the kinematic and dynamic free surface conditions. The conservation of F or the volume fraction of fluid is expressed as follows: ∂F ∂F =0 + Ui ∂xi ∂t
(2)
t + k
+
on the sediment phase, while the fluid flow remains uninfluenced. Such an assumption is accurate because: (i) as shown experimentally by Govan et al. (1989) and Yeganeh-Bakhtiary (1997) the presence of moving particles does not significantly modify the fluid flow at its upper part; (ii) within the hyper concentrated sediment layer, the velocity difference between the fluid flow and sediment particles is negligible. Therefore, the equations of motion of the ith particle on the vertically 2D coordinates are as follow:
24 CD = CD∞ + , Rep
CD∞ = 0.4
(11)
where s is the sediment density; A2 , A3 are the two and three dimensional geometrical coefficient of sediment; CM is the added mass coefficient (= 0.5); CD is the drag coefficient; CL is the lift coefficient (=4/3) [19]; dp is the sediment particle diameter; upi , wpi are the streamwise and vertical component of particle velocity; fn , fs are the normal and tangential components of the force acting between ith and jth particles; ωpi is the rotational angle of the ith particle; and Rep = (up dp /) is the sediment grain Reynolds number. In case of uniform particles the assessment of contacting particle is formulated by:
(xi − xj )2 + (yi − yj )2 ≤ dp
(12)
here (xi , xj ), (zi , zj ) are the coordinate of centroid of the ith and jth particles, dp is the particle diameter; and is the constant around 1.0. The acting force in normal and tangential direction between two contacting particles ith and jth can be estimated by the following relations: fn (t) = en (t) + dn (t)
(13)
fs (t) = es (t) + ds (t)
(14)
en (t) = min[en (t − t) + kn n , en max ]
(15)
es (t) = min[es (t − t) + ks s , es max ]
(16)
dn (t) = −n · n
(17)
ds (t) = −s · s
(18)
s = − n −
d t(ωpi + ωpj ) × n 2
(19)
where en , es are the forces working on the spring; dn , ds are the damping forces acting on the dashpot; n , s are the displacement of particles during the time step of the calculation t; kn , ks are the spring constants; n , s are the damping coefficient. In this model sediment grains were assumed to be non-cohesive
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99
Turbulent intensities are estimated according to Nezu [21] as follow:
u¯ 2 = 1.108 ×
¯ 2 = 0.336 × w
k
(27)
k
(28)
¯ are averaged turbulence velocities respectively in here u¯ and w horizontal and vertical direction. 3.2. Sediment phase domain Fig. 3. Numerical model configuration of Xie’s experiment [3].
grains, therefore the joint between contacting particles did not resist against the tension forces. On the other hand, a shear stress limit was utilized in the local tangential direction, the joint slips by exceeding this limit. The joint characteristics can be defined as: fn (t) = fs (t) = 0 when nen (t) < 0
(20)
|fs (t)| = |en (t)| when |es (t)| > |es (t)|
(21)
here is the coefficient of friction (= 0.55). The time step t should be proportional to critical time step or tc to satisfy DEM model stability. For a single mass-spring system with a single degree of freedom based on the mass m, t and tc can be estimated as:
t = tc /20;
tc = 2
m m ⇒ kn = 2 2kn
2 2 tc
(22)
where m is the mass of particle. In this model, the time step is adopted as t = 0.0002 s; thus kn and ks were estimated by Eq. (22) and the theory of elasticity as below: ks =
kn 2(1 + )
(23)
here (= 0.3) is the Poisson’s ratio. The damping coefficients were estimated from the critical damping conditions:
n = ˛cn 2
mkn ;
s =
n 2(1 + )
(24)
here ˛cn (= 1.0) was a calibrating coefficient. For more information please refer to Yeganeh-Bakhtiary et al. [14]. 3. Numerical method 3.1. Fluid phase
In addition to the fluid field, the sediment phase domain has to be specified. Before the main calculations, the packing calculation was executed to determine the initial position of particles. During this process, velocity of falling particles was monitored to assess the convergence of packing calculation. Fig. 4 shows a schematically sketch of the sediment domain placed in flow domain in front of a vertical breakwater. For defining the sediment motion, in fact two different domains are set together: an Eulerian domain for the fluid phase and a Lagrangian one for the sediments. The fluid domain is penetrated to the sediment domain. Thus, the fluid phase at its bottom grids is assumed to be containing with the sediment particles. The sediment transport occurs due to the interaction forces between the carrier fluid and sediment bed. In other words, the sediment particles is able to move freely in the Eulerian grids and the surrounding fluid velocity for each particle can be computed based on the particle location in the Eulerian grids. In the sediment phase the packing is essential to determine the initial position of sediment particles before starting calculations; thus two different sets of particles in the packing procedure was defined, namely main sediment particles and solid wall particles. The main particles were initially arranged with leaving a 0.001dp gap between each other. Prior to the main calculation phase, a preliminary packing procedure was executed to determine the locations of sediment particles. In other word, the sediment particles were subjected to gravity force in packing process to fill initial porosity leading to a more natural arrangement of bed particles. During the packing process, the velocity of moving particles was monitored to assess the convergence of packing calculation. At wall boundaries, mechanical characteristics of particles were set twice as the main particles in the domain; since these particles were set to have no displacement during interparticle contacts, therefore static contact characteristics were replaced with kinetic ones. knwall = 2kn ;
Fig. 3 gives a sketch of the fluid-phase domain for the configuration of Xie [3] experiment. The initial and boundary condition is quite similar to the last model of Yeganeh-Bakhtiary et al. [16] for the standing wave formation in front of a vertical breakwater. To be more concise herewith only the numerical wave generation in the flow domain is described. At the inlet boundary, the weakly reflective boundary condition, recommended by Petit et al. [20] was used for generating the incident waves and simultaneously absorbing the reflected waves from the vertical breakwater to prohibit the collision (intermixing) of the generated and reflected waves that propagate in the direction opposite the zone of interest, as follow: ∂Rr ∂Rr + Cr =0 ∂t ∂x
(25)
Rr = Rt − Rin
(26)
here Cr is the celerity of reflected wave; Rr , Rt and Rin are the variable associated with the reflected, computed and the theoretical wave values represents velocity, pressure and free surface displacement and Rin was calculated by the Stokes second order theory.
kswall = 2ks ;
wall = 2
wall = 2n ; n
wall = 2s ; s (29)
In every 0.1 s, the fluid velocities were fed to the sediment phase model, during each fluid phase time increment, the fluid velocity was considered to be constant and sediment particles were moving under action of a semi-steady flow. For every single particle, the flow velocities were calculated by locating the particle in the flow model mesh cells, then the orbital and the turbulence velocities could be estimated by interpolation and the lift and drag forces were calculated then. 4. Results and discussion 4.1. Model validation To assess the numerical model accuracy with one-way coupling approach, the model should be validated in the simple case studies. As mentioned before, the developed two-phase model consists of two parts; namely fluid flow model and sediment phase
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Fig. 4. Sediment domain placed in flow domain.
Müller experimental formula [23] to estimate the sediment transport described below: qb
q∗b =
∗ =
u2∗ ((s /) − 1)gd
((s /) − 1)gd3
∗ n = a( ∗ − cr )
(30)
(31)
here qb is the sediment transport rate, a and n are the empirical ∗ is the criticonstants; * is the Shields parameter; and cr cal Shields number. Based on the modified Shields diagram
Fig. 5. Computational domain for simulation of sediment transport under action of unidirectional flow.
model. The fluid flow model was extensively validated by YeganehBakhtiary et al. [16] for the simulation of fully and partially standing wave hydrodynamics in front of a vertical breakwater in terms of the mean and turbulent flow characteristics. Here the two-phase model was validated against two different hydrodynamics conditions, namely the sediment transport rate induced by unidirectional flow condition, and the standing wave-induced scour in front of vertical breakwater. For the former case, the sediment transport rate under the steady unidirectional flow was simulated and the results were compared with Meyer-Peter and Müller [23] experimental formula. The latter case, the wave-induced scour in front of vertical breakwater was simulated and were compared with experimental data of Xie [3]. In both cases the comparisons show a good agreement between the numerical results and experimental formula/data. Fig. 5 shows the numerical computational domain for simulation of sediment transport under action of steady unidirectional flow. In this simulation the sediment domain consist of uniform spheres, which were subjected to three different unidirectional flow with three different Shields parameter equal to 0.2, 0.4 and 0.8. Table 2 summarizes the physical characteristics of these three test cases. The results then were compared with Meyer-Peter and
Fig. 6. Sediment transport rate versus Shields parameter.
Table 2 Physical characteristics for simulation of transport under unidirectional flow. Test no.
s (kg/m3 )
D (mm)
U (m/s)
*
u*
1 2 3
2650 2650 2650
0.12 0.12 0.12
0.3 0.3 0.3
0.2 0.4 0.8
0.019 0.027 0.038
Fig. 7. Steady streaming pattern induced by standing waves in front of vertical breakwater.
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Table 3 Physical characteristics of Xie [3] experiment. H (m)
T (s)
h (m)
L (m)
H/gT2
d/T2
D50 (mm)
Sand type
Smax (cm)
0.05
1.17
0.3
1.71
0.0017
0.010
0.106
Relatively fine
1.3
recommended by Madsen and Grant [24], the critical Shields parameter is set to cr = 0.055, and the conventional values of a = 8.5 and n = 1.5 were used for the Meyer-Peter and Müller formula [23]. However, the modified constants as a = 7.5 and n = 1.1
were also adapted to the numerical results at higher values of shear stress (* > 0.4). Fig. 6 confirms good agreement between numerical results from present model and the bed-load transport formula.
Fig. 8. Development of scour/deposition in front of vertical breakwater under action of standing waves.
Fig. 9. (A) Wave induced scour pattern in front of vertical breakwater by Xie [3] and (B) comparison between model and experimental result by Xie [3] (diamonds).
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Fig. 10. Particle movement during a wave period in the zone between L/8 and L/4.
For the second case, the numerical two-phase model has been compared with the experimental results reported by Xie [3], which has been used as a benchmark case for evaluating scour computation induced by standing wave in front of vertical breakwater. Xie’s experiment was conducted in a 38 m long, 0.8 m wide and 0.6 m deep wave flume. At the beginning of the flume, the water depth was equal to 0.45 m; it then reduced to 0.3 m at the flat bed near the vertical breakwater with a 1:30 slope. Wave height and period were, respectively, 0.05 m and 1.17 s. The sediment mean diameter (dp50 ) was 0.106 mm and s = 2650 kg/m3 ; in this test the sand grains were relatively fine and suspension mode was
observed. In the numerical simulation, to reduce the computation time, the domain length was shortened to 14.5 m, which is discrete by 1 cm × 2 cm rectangular meshes. For the same reason, particles diameter was adjusted to 5.0 mm. To avoid any conflict between the experiment and the numerical simulation, Shields parameter in the numerical simulation was equal to the experiment. Table 3 presents the numerical configuration and summaries the physical characteristics of the Xie [3] experiment. Numerical simulation of scour relies on accurate simulation of flow pattern. Hence, the flow model was initially investigated to examine its capability in the prediction of flow pattern in front of a
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vertical breakwater. The fluid model have been validated with the experimental data of Xie [3] (see [16]) for standing waves orbital velocity. Generally, there was good agreement between the model results and different experimental data sets. Fig. 7 plots the steady streaming pattern generated by standing waves in front of the vertical breakwater. The orbital velocity of water particles was averaged during a whole period after the standing waves developed. As can be seen, the figure clearly shows a regular system of the recirculating cells of steady streaming generated in front of the vertical wall. The recirculating cells consist of top and bottom cells; while the bottom cells were attributed to the formation of bottom boundary layer as already illustrated in Fig. 1. It has been conjectured that the local scour is directly tied up to the dynamics of recirculating cells system of the steady streaming formation in front of the vertical breakwater [22]. To provide a better understanding of local scour formation, snapshots of scour/deposition development in the area extended from the near side to the distance of L/2 from the breakwater is depicted in Fig. 8. Rather long time lapse among snapshots was chosen to show how the bed sediments kept their motion. As shown
Fig. 11. Different sediment transport modes.
Fig. 12. Orbital and averaged sediment particle velocity during a wave period.
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Fig. 13. Averaged particle velocity in zero flow velocity zone during a wave period.
in the figure, the bed sediments responded to the top recirculating cells and after few wave periods, two sand bars were formed. The former sand bar formed at the vicinity of breakwater and the latter one at the distance of L/2 from the breakwater; consequently, the scour hole shaped between two bars at about L/4 from the breakwater. By continuing the influence of standing waves, the scour hole was getting deepen and after about 12T, it nearly reached the maximum scour depth. Then the bed sediments movement was decreasing with time; hence, after about 24T the maximum scour depth was almost achieved. It is apparent that after 26T a semi-equilibrium condition was reached in which the bed sediment movement was not changing the scour/deposition pattern of bed and the maximum scour depth could be estimated accordingly. In Fig. 9 the numerical result on the scour profile was compared with that of the Xie [3] experimental data. As can be seen, the scour pattern estimated by the model agrees very well at both the close vicinity and the distance of L/4 from the breakwater, a slight deviation, however, from the experimental data could be observed at the distance of about L/2 away from the breakwater. In other words, the simulation result fits very well with the scour pattern obtained
from the experiment. The maximum scour depth in the selected test case was 1.3 cm; while, the calculated maximum scour depth was 1.35 cm agrees well with the experiment. Qualitative comparison between Figs. 2a and 9 reveals that the scour pattern is identical to relatively fine sand pattern one. Thus the model was successful in the calculation of bedform and maximum scour depth. One of the main advantages of a two-phase flow model is its capability in estimating the scour pattern, as well as the sediment transport hydrodynamic in details. 4.2. Sediment transport hydrodynamic The hydrodynamic of sediment transport induced by standing waves in front of vertical breakwater is discussed in this section, Fig. 10 shows the snapshots of particle movements during a wave period with the time interval of T/6 in a distance between L/8 and L/4 away from the breakwater, in which the most particle movement was observed in this zone. The fluid velocity and the horizontal velocity of moving sediment are presented by the vectors and different colors, respectively. As it is observed in the figure,
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the orbital velocity started to increase from zero to its maximum at the lee ward direction of vertical breakwater, but the bed sediments shows a slight time lag to start their motion and to accelerate. The particle velocity and number of moving particles start to increase simultaneously with time. At the beginning the bed sediments keep their motion concurrently in the hyper concentrated layer and saltating particles over it. With increasing the wave orbital velocity, some bed sediment take parted from the bed and started to suspend in the wave flow domain after one quarter of wave period. In the mid period, the wave orbital velocity changes its direction to the seaward then it took a while for the moving sediment to accelerate in reverse direction. A possible illustration of the various different sediment transport modes is given in Fig. 11. In lower regions of snapshots, where the particles mainly move in hyper concentrated layer, the presence of hyper-concentrated layers of particles prevents the active sediment motion. Thus, a rather slow pattern of motion is formed in this region and the vertical displacements of the sand grains are negligible. In upper regions of snapshots, in which the saltating particles ride over the protuberance of lower particles, the concentration of moving particles is relatively low. Therefore, the motion of individual particle in this region is less frequently affected by its neighboring particles, compared to what takes place in the hyper-concentrated layer. The topmost region contains the least particle concentration where the suspended particles move more freely than the saltating particles and rarely collide to each other. These particles have the highest velocity and respond to the top recirculating cells very actively. The predominant parameter affecting mode of sediment transport is the momentum transfer due to fluid–sediment interactions between phases and collision stresses in the sediment phase. In the hyper-concentrated layer the vertical motion of sediments is negligible since the high sediment concentration induces very frequent interparticle collisions; therefore, the interparticle stresses in this layer are preserved. By contrast, in the upper layers, the concentration is low and vertical velocity of sediments Therefore, in these regions, according to Gotoh et al. [10] and Yeganeh-Bakhtiary et al. [11], there will be two paths of vertical momentum transfer due to: (i) vertical mixing with fluid; and (ii) vertical motion of grains keeping interaction with surrounding fluid. Since the vertical motions of saltating and suspended grains are significantly active, the exchanged momentum is well-mixed between the upper and lower sediment layers. In other words, the interphase momentum transfer is promoted much because frequent interparticle collisions are less likely to take place; hence, the fluid–particle interaction is the predominant mechanism of transport at the upper regions. Fig. 12 illustrates the relative averaged horizontal velocity of sediments and the relative horizontal orbital velocity, U/Ufmax , versus the relative height, Z/dp , in the distance of L/8 away from the breakwater. In this figure, the height the zone between 8dp to 12dp represents the hyper-concentrated zone; between 12dp to 14dp is the saltation zone and upper than 14dp is the suspension zone. The panels show that by starting a wave period, it took a small time less than 0.2 s (≈ T/6) for sediments to accelerate and follow the flow movement. Nevertheless, when the flow direction changes the same situation was observed, for example in T/2 sediment velocity in the suspension zone was even against the flow velocity. The graphs also reveal that sediment velocity can be half of flow velocity in maximum. The interesting thing about these graphs is the region of zero flow velocity. Fig. 13 shows the zero flow velocity zones individually for each graph as illustrated in Fig. 12. This zero flow zone represent bed region which flow velocity does not interfere inside it; however, it is readily seen in the all graphs, the sediment particles have a small velocity. This velocity is not produced by fluid flow, but by the vertical momentum transform from the upper moving particles.
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Fig. 14. Suspension load ratio in one wave period.
Fig. 14 depicts the percentage of the suspension load to the total load of sediment transport and the horizontal orbital velocity in the distance of L/8 from the breakwater during a wave period, the orbital velocity was measured 5 cm over the bed. The graph seems to have two maximum points, one after the T/4 when the flow velocity is lee ward and the other before 3T/4 when the velocity is see ward. The maximum percentage of suspended load is about 20%. 5. Conclusion A two-phase flow model has been developed to analyze numerically standing waves induced scour in front of a vertical breakwater. This model was formulated for the two key physical processes involved: the recirculating steady streaming formation and the sediment transport. The flow model is based on the mass and momentum conservation equations, the VOF technique for tracking the free surface motion and a k–ε turbulence closure model for describing the generation and dissipation processes of turbulence. The sediment phase was extended based on the 2D version of MBS-3D model of Yeganeh-Bakhtiary et al. [14]. In summary, the simulation results agree well with the experimental study of Xie [3]. The model was successful to calculate maximum scour depth as well as bedform profile after the semi-equilibrium profile was obtained. Moreover, in selected experiment for the simulation, sediments were relatively fine and therefore sediment suspension should occur during the scour, the numerical simulation results showed that the dominant sediment transport mode is hyperconcentrated with suspension which confirms the reliability of the model in simulation of scour mechanism. According to the analysis of this study, the following conclusions are drawn: • Most of the numerical models available for simulating scour in front of vertical breakwaters are based on single-phase flow
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models. An Euler–Lagrange two-phase flow model provides a powerful simulation tool for studying the sediment transport and hydrodynamics in the scour process. • Sediment transport rate is unsteady and non-uniform during a wave period and scour process; however as time elapsing the bed sediment movement decrease and a semi-equilibrium stage of scour is reachable for scour in front of a vertical breakwater. • There is a time lag less than 0.2 s (≈ T/6) between flow velocity and sediment velocity mainly due to the sediment inertia. Maximum suspension rate occurs with a time lag after T/4 and 3T/4 of each wave period where the maximum orbital velocity occurs. The maximum suspension ratio is about 20%. • Particles in hyper-concentrated layer – where the flow velocity is negligible and the flow does not interfere – move under action of upper moving particles that vertically transform momentum to the lower particles. References [1] Carter TG, Liu LFP, Mei CC. Mass transport by waves and offshore sand bedforms. Journal of Waterway Harbors and Coastal Engineering 1973;99(WW2):165–84. [2] Mei CC. The applied dynamic of ocean surface waves. World Scientific; 1989, 740 pp. [3] Xie SL. Scouring pattern in front of vertical breakwaters and their influence on the stability of the foundation of the breakwaters. Delft: Delft University of Technology; 1981. p. 61. [4] Gao X, Inouchi K. The characteristics of scouring and deposition in front of vertical breakwaters by broken clapotis. Coastal Engineering Journal 1998;40.(1). [5] Lee KH, Mizutani N. Experimental study on scour occurring at a vertical impermeable submerged breakwater. Applied Ocean Research 2008;30(2):92–9. [6] Chen B. The numerical simulation of local scour in front of a vertical-wall breakwater. In: Proceedings of the conference of global Chinese scholars on hydrodynamics. 2006. p. 134–8. [7] Gislason K, Fredsoe J, Sumer BM. Flow under standing waves Part 2. Scour and deposition in front of breakwaters. Coastal Engineering 2009;56(3):363–70. [8] Wiberg PL, Smith JD. Model for calculating bed-load transport of sediment. Journal of Hydraulic Engineering, ASCE 1989;115(1):101–23.
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