- Email: [email protected]
Vortex-induced suspension of sediment in the surf zone
Accepted Manuscript
Vortex-induced suspension of sediment in the surf zone Junichi Otsuka, Ayumi Saruwatari, Yasunori Watanabe PII: DOI: Reference:
S0309-1708(17)30268-3 10.1016/j.advwatres.2017.08.021 ADWR 2948
To appear in:
Advances in Water Resources
Received date: Revised date: Accepted date:
16 March 2017 28 July 2017 2 August 2017
Please cite this article as: Junichi Otsuka, Ayumi Saruwatari, Yasunori Watanabe, Vortexinduced suspension of sediment in the surf zone, Advances in Water Resources (2017), doi: 10.1016/j.advwatres.2017.08.021
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights • A major mechanism of sediment suspension by organized vortices produced under violent breaking waves in the surf zone was identified.
CR IP T
• Effect of the vortex-induced flows was incorporated into a suspension model.
AC
CE
PT
ED
M
AN US
• The model proposed reasonably predicts the measured sediment concentration due to violent plunging waves and significantly improves the underprediction of the concentration produced by previous models.
1
ACCEPTED MANUSCRIPT
Vortex-induced suspension of sediment in the surf zone Junichi Otsukaa , Ayumi Saruwatarib , Yasunori Watanabeb,∗ a Civil
CR IP T
Engineering Research Institute for Cold Region, Hiragishi 1-3-1-34, Sapporo 062 8602, Japan b Hokkaido University, Graduate School of Engineering, North 13 West 8, Sapporo 060 8628, Japan
Abstract
ED
M
AN US
A major mechanism of sediment suspension by organized vortices produced under violent breaking waves in the surf zone was identified through physical and computational experiments. Counter-rotating flows within obliquely descending eddies produced between adjacent primary roller vortices induce transverse convergent near-bed flows, driving bed load transport to form regular patterns of transverse depositions. The deposited sediment is then rapidly ejected by upward carrier flows induced between the vortices. This mechanism of vortexinduced suspension is supported by experimental evidence that coherent sediment clouds are ejected where the obliquely descending eddies reach the sea bed after the breaking wave front has passed. In addition to the effects of settling and turbulent diffusion caused by breaking waves, the effect of the vortexinduced flows was incorporated into a suspension model on the basis of vorticity dynamics and parametric characteristics of transverse flows in breaking waves. The model proposed here reasonably predicts an exponential attenuation of the measured sediment concentration due to violent plunging waves and significantly improves the underprediction of the concentration produced by previous models.
PT
Key words: sediment suspension, counter-rotating vortices, breaking waves 2016 MSC: 00-01, 99-00
CE
1. Introduction
AC
In the surf zone, a shoaling wave crest projects forward as an overturning jet and splashes onto the forward water surface [1]. The rotational fluid motion within the jet surrounding the air tube governs the flow involving a primary roller vortex at an early stage of wave breaking (see FIG. 1). Secondary jets are sequentially ejected to produce another roller vortex in the repeating splashing process, resulting in an array of horizontal roller vortices in the transition
5
∗ Corresponding
author
Preprint submitted to Advances in Water Resources
September 21, 2017
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 1: Schematic illustration of evolution of breaking waves; plunging breaker (top) and spilling breaker (bottom).
15
M
ED
10
region. The energetic splashing waves then culminate in turbulent bores propagating toward the shore in a bore region [2]. Nadaoka et al. [3] observed the formation of three-dimensional vortex structures involving obliquely descending eddies (ODEs) that developed along the principal axis of fluid strain, behind the rollers. They suggested that the ODEs contribute to the sediment suspension observed in the surf zone. This study focuses on mechanical roles of the breaking-wave-induced vortices to the sediment suspension and prediction of the sediment concentration in the surf zone.
FIG. 2 shows typical sequential images of experimental breaking waves at the plunging location over the sand bed (see also supplementary movie 1). After the wave front passes, FIG. 2(a), sediment begins to be picked up on the bed at the rear end of the aerated area, where the ODEs may entrap air bubbles, in FIG. 2 (b). The sediment is then rapidly transported upward with a rotating motion (FIG. 2 (c), and see also supplementary movie 1) and ejected up to the surface at (d). Watanabe et al. [4] identified the mechanism that forms the ODEs during the wave-breaking process; that is, initial primary vorticity changes in orientation in a stagnation-point flow induced between the adjacent roller vortices, and the re-oriented vortex filaments are stretched in the direction of the maximum eigenvector of fluid strain, resulting in a transverse array of counter-rotating vortices (that are identical to ODEs). When counter-rotating vortices approach a free-surface, the surface above the vortices is entrained into the inner water mass and is thus depressed, forming so-called scars (Sarplaya and Suthon[6],
AC
CE
20
PT
1.1. Obliquely Descending Eddies
25
30
3
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 2: Sequential images of experimental breaking waves passing over a sand layer in a wave flume shown in FIG. 3 (time interval of 0.2 s); (a) wave plunging results in the formation of secondary splash-up jets and entraps a rotating air tube in the inner water mass. (b) The ODEs, containing air bubbles, reach the bed and begin to pickup local sediments after passing the wave front (see arrow). (c, d) The suspended sediments are transported upward with a rotating motion. See also supplementary movie 1.
4
ACCEPTED MANUSCRIPT
45
CR IP T
40
AN US
35
Sarpkaya[5], Brocchini and Paregrine[7]). In the case of breaking waves, the subsurface counter-rotating vortices induce convergent flows to entrain the surface to bulk water and create the scars behind the wave crest (Watanabe and Mori [8]), and the development of scars on the jet surfaces penetrates the jet, fragmenting them into finger jets (Saruwatari et al. [9]). In addition to these contributions of breaking-wave-induced vortices on the surface boundary, several recent studies demonstrated explicit roles of the counter-rotating vortices (or ODEs) in the dynamics of sediment-laden flows near the bottom boundary. LeClaire and Ting [10] found high correlations between the population of suspended sediment particles and vorticity when the sediment was observed to be trapped in counter-rotating vortices. Zhou and Hsu [11] computationally identified that upward and downward vertical velocities caused by the penetration of ODEs in a near-bed region enhance sediment suspension next to the edge of the ODEs. According to Otsuka and Watanabe [12], regular patterns of divergent and convergent transverse flows appear in an array of counter-rotating vortices near the bed under breaking waves, suggesting that corresponding upward and downward vertical flows are also induced between the vortices. 1.2. Rouse distribution of sediment concentration
A general description of the mean concentration of suspended sediment, hci, has been provided by Rouse [13]. In an equilibrium state that the rate of settling equals the rate at which the sediment is lifted by turbulent diffusion,
M
50
wf hci = −hs i
(1)
ED
where η is the vertical coordinate with the origin at the undisturbed bed and wf is the settling velocity. h i indicates the temporal mean quantity. The sediment diffusion, hs i, is generally related to the turbulent diffusion of fluid, hf i, in terms of Schmidt number Sc as hs i = hf i/Sc . In steady open√channel flow, 1 τ0 assuming the logarithmic regime of the near bed flow du dη = κ η , the wellknown Rouse distribution [13] is given as the solution of equation (1), assuming Sc = 1; R η0 (h − η) hci = , (2) c0 η (h − η0 )
CE
PT
55
dhci , dη
where c0 is the reference concentration at the reference level η0 , h is the water p depth, R = wf /κ τ0 /ρ is the so-called Rouse parameter, κ is the von K´ arm´ an’s constant, and τ0 is the wall shear stress. In this case, the diffusion of fluid momentum takes a form of parabolic distribution; η κu0 , (3) hf i = η 1 − d p where u0 = τ0 /ρ is the friction velocity. In general, since Sc depends on c/c0 , hs i is also described as a function of the relative concentration c/c0 [14, 15].
AC
60
65
5
ACCEPTED MANUSCRIPT
1.3. Sediment concentration in the surf zone
hci = c0 e−wf η/hs i ,
80
CE
95
PT
90
ED
M
85
AC
100
105
(4)
which is the solution of equation (1) assuming the constant hs i. The identical profile of the concentration has been observed in the surf zone [19, 22], indicating the constant near-bed sediment mixing reasonably defines the breakingwave-induced turbulence, rather than the parabolic mixing (3) associated with logarithmic turbulent boundary layer. In model analyses of the suspended coastal sediment, the sediment mixing has also been studied for non-breaking waves (Lundgren [16], Jonsson [17], van Rijn [18]), and for vortex ripples under waves (Nielsen [19, 20]). Van Rijn [18] proposed a three-layer model of wave-induced sediment concentration. Larson and Kraus [21] modeled the vertical attenuation rate for an exponential concentration profile of the suspended sediment in the surf zone. Wang [22] compared the vertical profiles of the sediment concentration predicted by these models with experimental results in the surf zone, and found that both predictions were significantly lower than the measured results, regardless of breakers. Goda [23] integrated the previous datasets of the sediment concentration observed in the surf zones, leading to an empirical formula for sediment pickup rates. The remarkable variabilities in the observed concentration from the formula (0.18 – 4.2 times disparity), reported by Goda [23], suggest that uncertainties in predicting the suspension processes in complex surf zone flows remain. All these previous models exclude explicit effects associated with breaking-wave-induced vortices to carry sediment and maintain suspension against gravity, observed in FIG. 2, which may be one of the major factors that needs to be identified to improve the prediction of the sediment concentration in breaking waves. A recent progress of computational multiphase flow modeling, especially in a Lagrangian particle method[24] and turbulence-sediment coupled approach[25], was discussed at THESIS-2016 (the 3rd symposium on two-phase modeling for sediment dynamics in geophysical flows). While new numerical models are expected for further understanding of the sediment transport in the surf zone, the current study focuses on semi-analytical improvement of the previous empirical models in terms of vorticity dynamics in the surf zone. In this paper, fundamental statistical features of the suspended sediment in turbulent flows under breaking waves are experimentally investigated in §2. The mechanical effects of the counter-rotating vortices produced during the wavebreaking process to sediment transport are discussed through computational experiments using a Large Eddy Simulation (LES) of breaking waves in §3. In §4, a novel model of the vortex-induced vertical flow driving suspension was then developed based on the experimental and computational findings. The
AN US
75
CR IP T
70
Since unsteady, oscillatory boundary layer flows are produced in shallow water wave field, the velocity profile of the steady turbulent boundary layer flow, used in §1.2, is no longer acceptable. Bosman and Steetzel[27] found that the experimental mean concentration under coastal waves significantly deviates from the Rouse distribution (2) , and it follows the exponential distribution;
6
ACCEPTED MANUSCRIPT
110
reliability of the proposed model is discussed through comparisons with the experimental results as well as with the predictions of the previous models. Our findings and conclusions are summarized in §5.
CR IP T
2. Laboratory Experiment 2.1. Experimental setup
130
135
AN US
AC
CE
140
M
125
ED
120
The experiments were performed in a 24-m long wave flume, 0.4 m wide and 1.0 m deep, fitted with a piston type wave maker and active wave absorption (see FIG. 3 a). A sand layer was installed in a rectangular sand pit (3 m long, 0.4 m wide, and 0.1 m deep) to cover the surf zone of the 1/20 slope beach. The median diameter, d50 , of the sand was 0.20 mm. We defined the coordinate system for the experiment with the origin at the point of wave-breaking in the horizontal shoreward x axis and at the still water level in the vertical z axis (see FIG. 3 b). The surface elevation ζ(x, t) was defined as the vertical displacement of the free surface about the still water level, z = 0. The sediment concentration, fluid velocity, and surface elevation were measured at the same cross-shore locations that were traversed at 100-mm intervals from the breaking point (x = 0) in the x direction over the surf zone. The concentration of suspended sediment was measured using an optical concentration sensor (PMT5-50, Kenek) that was vertically traversed at 10-mm intervals from the bottom to the level of the wave trough. Particle Imaging Velocimetry (PIV) estimates planar distributions of the fluid velocity from sequential images of tracers recorded at the measurement sites (see FIG. 3 b and c). A continuous wave YAG laser sheet (DPGL-8W, Japan Laser), which was guided into the water through an optical fiber, was emitted shoreward from the offshore side of the breaking point to illuminate the tracer particles in a horizontal-vertical cross-section for recording by an 8-bit high-speed video camera (FASTCAMSA3, Photron) (see FIG. 3 b and c). The field-of-view of the camera was 120 mm× 60 mm. Neutral buoyant fluorescent particles approximately 20 µm in diameter were used as tracers of the fluid flow. Laser light (with a wavelength of 532 nm) was reflected by the suspended sand, entrained air bubbles, and the free-surface of the waves. A camera equipped with a high-pass optical filter (cut-off wavelength < 580 nm) prevented the recording of reflected light, and captured only the laser-induced fluorescent light (with a light spectrum peak of 650 nm) emitted from the fluorescent tracers. Eight-bit, 1024×512 pixel images were recorded at 250 Hz with an exposure time of 1/1500 s and stored on a PC connected to the camera as uncompressed bitmaps. A capacitance-type wave gage was used to record the simultaneous surface elevation at the measurement sites. Analog signals transmitted from the wave gages and concentration sensor were converted to digital data with a sampling frequency of 100 Hz and saved on a PC. When the first wave passed the wave gage installed offshore (see FIG. 3 a), a trigger signal was sent to the high-speed video camera and PC to simultaneously start recording the images and surface elevation data. This synchronized the wave phase of the 30 trials performed for each experimental case to allow
PT
115
145
150
7
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 3: Experimental setup; (a) wave flume, (b) cross-shore locations of measurements and definitions of the coordinate system and surface elevation ζ, (c) photograph of the measurement site (high-speed camera records red fluorescent light emission from tracers on a green laser sheet).
8
ACCEPTED MANUSCRIPT
Table 1: Wave conditions
T (s)
1 2 3
1.4 1.8 2.0
*1
Hb (cm)*2
hb (cm)
12.5 14.7 13.7
13.8 15.0 14.1
*3
*4
bed slope
ξ
1/20 1/20 1/20
0.247 0.293 0.337
*1
Wave period Breaking-wave height *3 Breaking-water depth *4 Surf similarity parameter *2
breaker type Spilling Plunging Plunging
CR IP T
Case
Table 2: Characteristic location and depth of breaking waves in the surf zone (see FIG. 1).
3
Bore Region
(xp ≤ x < xB )
(xB ≤ x)
0.0
–
13.8
–
0.0
60.0
15.0
12.0
0.0 14.1
0.0 – 122.0
122.0 – 276.0
13.8–7.7
7.7 –0.0
0.0 – 110.0
110.0 – 300.0
15.0–9.5
9.5 –0.0
68.0
0.0 – 124.0
124.0 – 282.0
10.7
14.1–7.9
7.9 –0.0
PT
for statistical analysis of the results on the basis of ensemble averaging. The sand bed surface was flattened before each trial to achieve identical initial bed conditions for every trial. We performed three experimental cases of wave breaking (see Table 1). The wave-breaking process depends on the breaker type; plunging jets are sequentially splashed to produce multiple rollers in a plunging breaker, as illustrated in FIG. 1, while a smaller roller vortex produced at the wave crest spilled down on the wave face in a spilling breaker. Table 2 summarizes the experimental locations characterizing the surf zone flows; the wave plunging point (xp ), transition region (xp ≤ x ≤ xB ) and the bore region (xB ≤ x) (see also FIG. 1). The breaker type is commonly characterized by a surf similarity parameter [30] defined by p (5) ξ = tan θ/ Hb /L0 ,
CE
155
Transition Region
(xp )
AN US
2
x (cm) h (cm) x (cm) h (cm) x (cm) h (cm)
Plunging Point
M
1
Breaking Point
ED
Case
AC
160
165
where θ is the bottom slope angle and L0 is the offshore wavelength. The breaker type and ξ for each case are summarized in Table 1.
9
ACCEPTED MANUSCRIPT
180
CR IP T
175
AN US
170
The acquired images were calibrated with real measurements at a 0.1-mm/pixel resolution. The image noise was reduced using a median digital filter. In the PIV analysis, a planar distribution of the instantaneous fluid velocity was estimated on the basis of a standard cross-correlation method [31] for 31 × 31 pixels of interrogation windows on two sequential images. The ensemble average was taken over 30 trials to definethe ensemble averaged velocity u and the turbu1 lent energy k = 2 u02 + w02 , where the velocity fluctuation u0 = (u0 , w0 ) was defined as the deviation from u through Reynolds decomposition. We found that the laser light transmission was interrupted in highly aerated areas and thus significant errors occurred there. The unreliable results contaminated by the aeration were eliminated from the statistics when the maximum correlation between particle image windows is less than 0.65, and/or |u − um | > 1.5σ, where um and σ are the mean velocity and standard deviation over the neighboring window locations, respectively. When available trials were less than 30, the measured data was not used for the analysis to ensure statistical reliability of the current results. 2.2. Experimental results
M
185
FIG. 4 shows the ensemble-averaged surface elevation, ζ, fluid velocity, u = (u, w), turbulent energy, k, sediment concentration, c, and the cross-shore distribution of u, k and c averaged over a FOV, for the spilling wave case, when the maximum sediment concentration is achieved. Readers can also see the sequential results throughout the breaking process in supplementary movie 2. While the maximum shoreward velocity appeared behind the wave crest in x ∼ 100 cm, higher turbulent energy extended over a wide region under the wave, 60 ≤ x ≤ 120 cm. The sediment concentration became maximal at a location more distant from the wave crest, x ∼ 40 cm (see FIG. 4 d), indicating no direct correlation of the concentration with the spatial development of shoreward flow and turbulence. In plunging wave cases (see FIG. 5 and FIG. 6), the turbulent energy was locally intensified through repeating splash-up processes (see FIG. 1), transported downward, and widely distributed behind the wave front (see also supplementary movies 3 and 4). We found that a moderate level of turbulent energy (20 – 50 % of the observed maximum turbulent energy) was transported to the bottom, which may affect the diffusion of the suspended sediment. While the sediment concentration extended over a wide region (0 ≤ x ≤ 90 cm for case 2, and 60 ≤ x ≤ 100 cm for case 3), this region does not coincide with where the energetic turbulence appeared (40 ≤ x ≤ 110 cm for case 2, and 60 ≤ x ≤ 180 cm for case 3) nor where shoreward fluid velocity occurred under the wave crest (90 ≤ x ≤ 120 cm for case 2, and 110 ≤ x ≤ 160 cm for case 3). The high concentration was locally maintained over the depth in the duration of t/T ∼ 0.3 − 0.4 after passing the upstream end of the region containing energetic turbulence in both cases (see supplementary movies 3 and 4). These results suggest another mechanism to uplift the suspended sediment and increase the concentration at higher levels in addition to effects of
CE
195
PT
ED
190
AC
200
205
10
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 4: Ensemble-averaged surface elevation (a), velocity vector and contour of turbulent energy (b), sediment concentration (c), and cross-shore distributions of k, c and u averaged over a FOV (d) at the phase in which the maximum concentration is achieved, in case 1 (spilling breaker). See also supplementary movie 2.
11
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 5: Ensemble-averaged surface elevation (a), velocity vector and contour of turbulent energy (b), sediment concentration (c), and cross-shore distributions of k, c and u averaged over a FOV (d) at the phase in which the maximum concentration is achieved, in case 2 (plunging breaker). See also supplementary movie 3.
12
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 6: Ensemble-averaged surface elevation (a), velocity vector and contour of turbulent energy (b), sediment concentration (c), and cross-shore distributions of k, c and u averaged over a FOV (d) at the phase in which the maximum concentration is achieved, in case 3 (plunging breaker). See also supplementary movie 4.
13
ACCEPTED MANUSCRIPT
Table 3: Experimental data of the reference concentrations and attenuation rates.
case 2
100
120
140
-7.34 -8.55 -5.38 897.5 1147.6 646.7 -10.10 -10.68 -9.20 1631.4 1968.5 1289.7 -4.40 -5.14 -3.28 1128.3 1404.3 853.5
-11.01 -12.17 -8.81 867.0 1162.1 577.1 -9.50 -10.73 -7.73 1614.1 1995.5 1235.3 -9.31 -9.79 -8.52 1532.1 1956.8 1107.8
-13.01 -14.71 -9.25 739.5 1019.3 452.2 -8.35 -7.92 -9.08 1273.7 1557.5 989.8 -12.52 -11.56 -14.80 2029.1 2811.8 1239.9
-11.10 -13.58 -4.63 572.2 815.8 312.3 -9.52 -7.33 -13.98 881.6 1098.2 672.5 -18.91 -19.68 -17.15 1937.5 2727.3 1143.8
-25.57 -25.22 -26.23 657.0 851.1 463.0 -12.00 -14.40 -7.56 622.3 858.2 387.8 -16.37 -17.64 -14.13 1386.8 1899.3 877.7
-22.70 -26.43 -15.53 458.8 657.5 265.3 -13.94 -15.98 -10.29 555.0 756.0 356.0 -26.62 -30.76 -18.42 1328.2 1948.0 728.7
the turbulence caused by wave breaking and the friction in the wave boundary layer. The vertical profile of temporal mean sediment concentration, hci, was conventionally estimated by solving the one-dimensional convection-diffusion equation derived from the continuity equation for sediment, equation (1). As already explained in §1.3, the well-known Rouse distribution of hci for steady turbulent boundary layer flow, equation (2), provide an inaccurate approximation of the one observed in wave field where the sediment is periodically driven by unsteady, oscillatory boundary layer flows over the seabed [27]. In the surf zone where the breaking-wave-induced turbulence and vortices govern the near-bed sediment mixing, the concentration has been observed to follow the exponential distribution (4) (e.g. Nielsen [19], Wang et al. [22]). A number of models for the reference concentration, c0 , and attenuation rate, hai, for the general form of the exponential distribution,
CE
PT
215
80
ED
210
60
M
case 3
hai (m ) hai+ (m−1 ) hai− (m−1 ) c0 (ppm) c0+ (ppm) c0− (ppm) hai (m−1 ) hai+ (m−1 ) hai− (m−1 ) c0 (ppm) c0+ (ppm) c0− (ppm) a (m−1 ) hai+ (m−1 ) hai− (m−1 ) c0 (ppm) c0+ (ppm) c0− (ppm)
40
CR IP T
case 1
−1
AN US
x (cm)
AC
220
225
hci = c0 ehaiη ,
(6)
have been proposed for non-breaking water waves (Nielsen [20], van Rijn [18]) and breaking waves (Larson and Kraus [21]). According to van Rijn [18], the sediment concentration has different vertical profiles in a near-bed mixing layer, governed by turbulence and vortices, and above the layer with orbital fluid motion (see also Bosman and Steetzel [27]). FIG. 7 shows the vertical profiles of the measured time-averaged concentra14
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 7: Cross-shore variations in vertical profiles of the time-averaged sediment concentration (circle) with error bars and the exponential approximation curve (red line); case 1 (left), case 2 (middle), and case 3 (right). The dotted and broken lines represent the exponential approximation curves of the standard deviations above and bellow the mean concentration, respectively.
15
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ED
Figure 8: Cross-shore variations in the predicted attenuation rate by Larson and Kraus [21], Goda [23] and van Rijn [18]; case 1 (red), case 2 (blue), and case 3 (black).
PT
tions in the surf zone, together with the exponential curves, hai, approximated by the least squares method . The error bar represents the standard deviation of the mean concentration. The exponential approximation appropriately describes the vertical distribution of the mean concentration over the measured depth (below the wave trough level) . Van Rijn [18] suggested that the mixing effects in the surf zone can be represented by increasing the mixing layer thickness where the concentration is defined by the exponential distribution, which may support the current results of the exponential attenuation over the depth. We find the standard deviations of the mean concentration appear to be maximal in the near-bed region and decays away from the bed for any cases, suggesting higher uncertainty of the near-bed concentration, which is discussed in §4.5. The attenuation rates for the concentrations at the standard deviations above and bellow the mean concentration, hai+ and hai− , are also well approximated by the exponential curves. All estimated values of the attenuation rate, hai, the reference concentration, c0 , and those estimated from the deviation concentrations (hai+ , hai− , c0+ and c0− ) are summarized in table 3.
CE
230
AC
235
240
16
ACCEPTED MANUSCRIPT
Van Rijn [28] proposed the three-layer model of the sediment concentration. In the near-bed layer where the exponential form is assumed, the sediment mixing coefficient is modeled as hs,vr i = 0.004D∗ Uδ δs ,
250
where the particle parameter D∗ = d50 (s − 1) g/ν , Uδ is the peak orbital velocity, s is the specific gravity, ν is the dynamic viscosity, the thickness of the 0.5 sediment mixing layer δs = 0.3h (H/h) , h is the water depth, and H is the wave height. Van Rijn [18] suggested 0.05 m ≤ δs ≤ 0.2 m for breaking waves. Larson and Kraus [21] suggested the mixing coefficient hs,lk i = kd h (D/ρ)
1/3
(8)
where the energy dissipation rate D ≈ ∂F/∂x, F is the energy flux, and the dimensionless empirical coefficient kd ≈ 0.03. Goda [23] empirically integrated the observation datasets of the mean sediment concentration, and an empirical relation of the estimated attenuation rates was suggested in Goda [29]: A hag i = − D−0.2 , h
(9)
where A=6. The predicted attenuation rates for van Rijn [28], havr i = −wf /hs,vr i, for Larson and Kraus [21], halk i = −wf /hs,lk i, and for Goda [23], hag i, are shown in FIG. 8. All of these predictions exhibit much higher attenuations than the current experimental attenuations, hai, hai+ and hai− , throughout the surf zone, which may result in an underprediction of the concentration over the depth. Wang [22] came to the identical conclusions that both the predicted concentrations by van Rijn [18] and Larson and Kraus [21] are much lower than the measured concentrations. Van Rijn [18] also suggested a considerable underprediction of the concentration observed in a large-scale wave flume, while the model results were consistent with the measurements made in a small-scale wave flume. In the next section, the possible mechanical causes of the inconsistent predictions and the required parameters to improve the prediction are discussed through computational experiments of material transport in breaking-waveinduced turbulent flows.
CE
270
PT
265
ED
M
260
,
AN US
255
(7) 2 1/3
CR IP T
245
AC
3. Computations
275
280
Three-dimensional vortices are produced during the splash-up wave breaking process (Watanabe et al. [4], Saruwatari et al. [9], Lubin and Glockner [32]), and may enhance sediment suspension (Nadaoka et al. [3], LeClaire and Ting [10], Zhou and Hsu [11]). As the instantaneous structure of vortices causing local suspension is difficult to be experimentally quantified, we perform computational experiments with breaking waves to find the mechanisms of the near-bed material transport induced by the three-dimensional vortices. 17
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
PT
Figure 9: Computational domain.
3.1. Computational procedure
CE
3.1.1. Computing methods LES was performed to compute incompressible turbulent flows under breaking waves in three-dimensional space. The computational scheme is summarized in this section and the full details are described in Watanabe et al. [4, 33]. The filtered Navier-Stokes equation was used as the governing equation;
AC
285
290
˜ Du Dt
= −∇˜ p − ∇ · τ + ∇ · τ˜0 + g,
(10)
˜ represents the velocity, where the variables with a tilde denote filtered values, u p˜ is the pressure, τ is the sub-grid scale stress tensor, τ˜0 is the viscous stress ˜ · ∇ is the filtered material tensor, and g is the gravity vector. D/Dt = ∂/∂t + u derivative. The sub-grid scale viscosity model by Yakhot & Orszag [35] was
18
ACCEPTED MANUSCRIPT
Dφ Dt
305
(11)
A sharp interface model to fulfill normal and tangential dynamic conditions at the surface (Watanabe et al. [33]) was introduced in this study. In this model, the normal jump condition, which defines the pressure at the surface to balance with normal viscous stress and surface tension, was used to compute a Poisson equation for pressure. Since the tangential dynamic condition, imposing zero tangential shear at the free-surface, ensures generation of vorticity on a curved surface [34], the model provides reasonable estimates of the primary spanwise vorticity on highly curved inner surface of the overturning jet, and the following formation of finger jets via surface-vorticity interactions [9]. In the framework of a fractional step method, a discretized form of equation (10) was decomposed into discretized advection and non-advection equations. The Cubic Interpolation Polynomials method (Yabe and Aoki [37]) was used to solve the advection equation, while a predictor-corrector method was used to update the non-advection equation. A Poisson equation for pressure was iteratively solved using a multi-grid method. The advection equation for φ, equation (11), was also computed by the Cubic Interpolation Polynomials method. The current numerical technique was validated in Watanabe et al. [33], and the LES model for breaking-wave-induced turbulence was also validated in Watanabe et al. [4].
CE
320
3.1.2. Computational conditions LES was performed in a Cartesian staggered grid system, 1050×66×74 with a dimensional resolution of 6.0 mm per cell, in a rectangle domain tilted with an angle of 1/20 with respect to the horizontal axis x (see FIG. 9). The grid spacings in wall unit were (∆x∗ , ∆y ∗ , ∆z ∗ ) = (60.1, 61.5, 61.4), and an ODE was resolved by about 5.5 grids in the spanwise direction. The measures of the domain were identical with those of the sloping section of the laboratory wave flume (see FIG. 3 a). The analytical velocity, pressure, and surface elevation on the basis of the second-order cnoidal wave theory are given to generate regular progressive waves at the offshore boundary. The LES results depend on the turbulent boundary conditions. While a wall law based on a logarithmic velocity profile has been used for computations of turbulent boundary layer flows, a primitive non-slip condition has been applied to wave breaking flows [38, 39, 4, 40], since the breaking-wave-induced turbulence transported to the near-bed region disrupts the boundary layer flow and thus the logarithmic profile is no longer acceptable there. We also employed the non-slip impermeable boundary condition at the bottom. A periodic condition
PT
315
ED
M
310
0.
AN US
300
=
CR IP T
295
used to model the sub-grid stress τ . All variables are non-dimensionalized by the water depth, gravity, and water density. The level-set method (Osher and Fedkiw [36]) was introduced to compute the free-surface form. In this method, a level-set function φ, representing the signed distance from the surface, defined the surface location at φ(x, t) = 0, and followed the advection equation;
AC
325
330
19
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 10: Evolution of the breaking wave surface and iso-surface of the dye concentration (c∗ = 0.2), (left), and the structures of the vortex cores (right) at an early stage of wave-breaking in case 3; the vortex cores for primary rollers with spanwise vorticity are colored green, and those for vortex loops with a counter-rotating vorticity are colored red for positive streamwise vorticity and blue for negative streamwise vorticity. See also supplementary movie 5.
20
ACCEPTED MANUSCRIPT
355
360
CR IP T
c∗ = c∗0 ea
AC
CE
365
AN US
350
M
345
ED
340
was applied at the side boundaries. We confirmed that the wave field in the surf zone achieved a quasi-steady state after the third wave-breaking occurred from the initial still water state; that is, the breaking-wave height and depth identical to the experimental measurements (see Table 1) were attained for the third breaking waves. While the current single-phase (liquid phase) computation excludes the dynamics of the settling solid sediment, a fraction of the numerical dye was computed as the simplest model of the volume concentration of passive sediment, which may approximately describe dilute particle-laden flows containing very fine sediment without contact stress, characterized by a small Stokes number. It should be noted that Stokes number for the experimental conditions in §3, ρs d250 p g St = 18µ h ≈ 0.025 1, indicating passive particle behaviors to the carrier flows. The dye concentration, c∗ , was defined in 0 ≤ c∗ ≤ 1, and governed by the advection equation Dc∗ = 0. (12) Dt Since equation (12) excludes the effects of turbulent diffusion and settling, the dye is passively advected without approaching an equilibrium state that the settling sediment is kept in suspension by the turbulent diffusion. In general, the turbulent diffusivity of a solid particle, described in terms of Schmidt number (Sc ), varies in gravity environment due to inertia of the particle, depending on relative response time to turbulence time-scale [41, 42]. Sc has been empirically determined for trivial flows, such as Sc =0.7 [43] and 0.56 [44] for open channel flows, 0.8 for flow bellow a pipeline [45], and variable Sc (0.56 – 0.7) for oscillatory flows[15]. However no available Sc model has been proposed for the breaking-wave-induced turbulence with wide-ranging fluctuation timescales. Since a coherent formation of sediment clouds, observed in FIG. 2, may be caused by the advection transport by organized flows not simple diffusion process, we concentrated on computing the governing advection effect to the coherent suspension by equation (12) without any bias associated with uncertain Sc during the wave breaking process. In this computational test, the advection effect of vortices to the initial suspension behavior of near-bed sediment is considered as a deviation from the equilibrium concentration typically observed in the physical experiments. Analogous to equation (6), the initial exponential profile of the concentration was assumed to be in an equilibrium state;
PT
335
370
∗ ∗
z
,
(13)
where the reference concentration c∗0 =1, the attenuation rate a∗ ≈ −30, and z ∗ is the normal coordinate from the bottom boundary (see FIG. 9). This profile was initially given for the entire surf zone when the third wave commenced breaking. It should be noted that the organization of dye transport is independent on the initial c∗ profile since c∗ is passively transported by the local fluid flow, following equation (12). Therefore the dye quantity never affects the carrier flow. 21
ACCEPTED MANUSCRIPT
380
While the current single-phase LES model has been validated for breaking waves [4] and for liquid impacts to free-surface [9], the identical model without a sediment model was used in the computational test, assuming minor sediment effects at an early stage of the suspension process when the breaking-waveinduced turbulence arrives the near-bed region. The validity of the current model assumptions at the early suspension process is discussed in Appendix.
CR IP T
375
3.2. Vortex structures
390
CE
405
PT
400
ED
M
395
AN US
385
While vortex dynamics analogous to Watanabe et al. [4] were observed in the current computation, distinct features of the near-bed transport process were identified in the computations. FIG. 10 shows the evolution of the surface forms, the iso-surfaces of the dye concentration, and the distributions of the vortex cores defined by the λ2 method (Jeong and Hussain [46]), (see also supplementary movie 5). After wave plunging (FIG. 10 a), a primary horizontal roller vortex (colored green) is initially produced around the inner surface of an overturning jet. The vortex loops with streamwise counter-rotating vorticity (colored red and blue for positive and negative vorticity, respectively) are then formed between the preceding secondary jet and the roller vortex (FIG. 10 b). The upstream and downstream bends of the vortex loops are wrapped by each primary roller and thus stretched in the oblique direction to organize a rib structure of counter-rotating vortices between them (FIG. 10 c). Because the counter-rotating vortices entrain fluid downward from the free-surface into the inner water mass to form longitudinal scars along the vortex axes[6, 7], transverse surface deformation is observed behind the wave front (see supplementary movie 5, and also Watanabe and Mori [8]). We find that transverse fluctuations of the dye concentration also occur when the upstream bends of the counter-rotating vortices approach the bed. The fluctuations of the dye concentration are amplified according to the evolution of the three-dimensional vortex structures (see FIG. 11 and supplementary movie 5). Zhou et al.[11] computationally observed that velocity fluctuations caused by the ODEs are associated with sediment suspension on a barred beach. They found no suspension occurred between the cores of ODEs where the downward vertical velocity fluctuation exited, while the sediments were advected upward on the side of the ODE with upward velocity fluctuation. The current results that the dye suspensions locally occur at the transverse locations where the vortex loops are wrapped by the upstream primary roller (see FIG. 11 b and c) is consistent with the finding by Zhou et al.[11], which supports the current dye model describes major features of the initial suspension event govern by ODEs. Now we discuss how the orientation of the vertical flows alternates around ODEs and what is the mechanical role of ODEs to the near-bed sediment transport, in the following sections.
AC
410
415
3.3. Dye transport in vortex field FIG. 12 shows the contours of the streamwise vorticity and the dye concentrations on a (y ∗ - z ∗ ) cross-section at x∗ = 12.77 where the maximum rise
22
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 11: Evolution of the breaking wave surface and iso-surface of the dye concentration (c∗ = 0.2), (left), and the structures of the vortex cores (right) during the splash-up process in case 3; the vortex cores for primary rollers with spanwise vorticity are colored green, and these for vortex loops with counter-rotating vorticity are colored red for positive streamwise vorticity and blue for negative streamwise vorticity. The phases (a) - (c) and the rectangle cross-section at x∗ =12.77 correspond to FIG. 12. See also supplementary movie 5.
23
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 12: Contours of the streamwise vorticity (ωx ) and the dye concentration (c∗ ) on a transverse-vertical (y ∗ − z ∗ ) plane at x∗ = 12.77 (see the rectangle cross-section in FIG. 11). The phases (a) - (c) correspond to FIG. 11. See also supplementary movie 6
.
24
ACCEPTED MANUSCRIPT
430
435
CE
450
PT
445
ED
M
440
CR IP T
425
AN US
420
of dye was observed, at the corresponding phases in FIG. 11 (see also supplementary movie 6). The transverse wavy distributions of the dye concentration appear near the bed when the descending vortices approach the bed (see FIG. 12 a). The fluctuations are amplified beneath the counter-rotating vortices where indicated by black arrows in FIG. 12 b, and then the concentrated dye is vertically ejected between the vortices and extended up to the surface level (see FIG. 12 c and supplementary movie 6). This suspension process of the dye may be interpreted in terms of vorticity dynamics. The convergent near-bed flow is induced at the convergent points, indicated by black arrows in FIG. 12 b, beneath the vortex pair with clockwise (positive) vorticity located on the right side of the convergent point and counterclockwise (negative) vorticity on the left side (see also FIG. 15). This flow induces transverse dye transport toward the convergent point from both sides, resulting in regular patterns of depositions in the transverse direction (FIG. 12 a). The deposited dye is carried by upward flows induced between these vortices and ejected up to the free surfaces (see FIG. 12 b and also FIG. 15). In contrast, the counter-rotating vortices with the opposite vorticity orientation induce divergent near-bed flows to sweep the dye away from the divergent points (red arrows in FIG. 12). As the ejected dye is consecutively carried upward by the vortex-induced flows, a rapid increase in the dye concentration is observed over the depth (see supplementary movie 6). Although the computation shows the passive dye transport without settling effects, the identical contributions of the counter-rotating vortices to the initial suspension process may cause an analogous upward sediment transport, which is again discussed in §3.4. Once the sediment is suspended, the concentration may be modified by the settling effect, which may differ from the computed dye motion at a later stage. FIG. 13 compares the typical sequential images of experimental breaking waves, and the computed vortex cores and dye concentration at y ∗ = 1.2 where the maximum dye rise was observed (see FIG. 12). Assuming air bubbles are entrapped within the upper parts of ODEs during wave plunging (FIG. 13 a– b) and are transported to depths with the descending vortices (FIG. 13 c – e), the computed vortex cores present consistent displacements with the observed ODEs during passage of the breaking wave (FIG. 13 f – j). When the computed vortex cores arrive the seabed, the bottom layer with high dye concentration becomes thicker near the offshore ends of the cores (at x ≈ 60 – 65 cm). The upward ejection and overturning transport of the dye can be then observed near x ≈ 65 cm (corresponding to x∗ ≈ 12.77 in FIG. 11) where the spanwise roller vortex wraps rear ends of the streamwise vortices (FIG. 11) and the vertical dye ejection occurs between the vortices (FIG. 12). Munro et al. [47] experimentally studied sediment resuspension and erosion induced by a vortex ring interacting with a sediment layer. They found the near-bed sediment is displaced in a high shear region bellow the ring core, resulting in the formation of a circular crater mound with spokelike scars (azimuthal deformation) in the crater interior. The azimuthal array of the radial deposition and erosion was concluded to be caused by the local sediment displacement owing to the azimuthal velocity induced by the azimuthal instability
AC
455
460
25
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 13: Sequential snapshots of the experimental suspended sediment at the plunging point, (a) – (e), and the computed vortex cores and dye concentration at y = 1.23, where the maximum dye rise is achieved, at the corresponding phases (f ) – (j); the phase intervals of (a) – (d) and (e) – (h) are 0.156 s.
26
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
Figure 14: Schematic illustration of the sediment suspension by the vortex ring; the divergent and convergent near-bed flows induced by the counter-rotating azimuthal vortices above the bed may cause bed load sediment transport toward the converging locations where the sediment is carried by upward flows induced between a pair of the counter-rotating vortices. The suspended sediment may be then carried by the primary vortex flow and rolled up around the ring perimeter.
27
ACCEPTED MANUSCRIPT
475
480
M
485
CR IP T
470
of the vortex ring[48]. Watanabe et al. [33] computationally found that the azimuthal instability results in a wavy deformation of the primary vortex ring and the formation of counter-rotating radial vortices wrapped by the vortex ring. FIG. 14 illustrates the possible mechanism of azimuthal suspension and deposition by the unstable vortex ring wrapping counter-rotating vortices approaching the sediment bed. The counter-rotating radial vortices wrapped by the primary vortex ring may induce divergent and convergent azimuthal nearbed flows beneath each vortex pair along the ring axis, resulting in the azimuthal array of the spokelike sediment depositions and erosions in the crater. The suspended sediments are then carried by the primary vortex flow and are rolled up around the ring perimeter, resulting in coherent rolled-up sediment clouds toward the crater interior[47]. The suspension and deposition processes by the unstable vortex ring provide analogies with those by the breaking-wave-induced vortices. In the wave breaking case, the dye lifted between the counter-rotating vortices from the depositions (FIG. 12) are rolled up by the primary roller to form the overturning dye pattern around the roller vortex (FIG. 13 right). The similar patterns of rolled-up sediments are also observed in the experiments (see FIG. 13 (c) – (e), and also FIG. 2 (c) and (d)). The high sediment concentration in 60 ≤ x ≤ 100 cm, observed in FIG. 6, may be interpreted by the effects of the suspension around the primary horizontal roller wrapping the streamwise vortices via the combined mechanisms of the vortex induced flows by the counter-rotating vortices and the primary roller.
AN US
465
ED
CE
495
PT
490
3.4. Summary of computed results FIG. 15 schematically illustrates the mechanical contributions of the counterrotating vortices to the suspension process of the sediment. The counter-rotating vortices, produced between the primary rollers, are stretched obliquely downward, and are achieved at the sea bed. The convergent and divergent flows are induced over the bed underneath the counter-rotating vortices, resulting in the transverse bed-loads of sediment and thus the transverse patterns of local depositions at the convergent points on the bed. The deposited sediment is then carried by the upward flow induced between the counter-rotating vortices, causing local suspension, as observed in FIG. 2 and FIG. 13. Since the suspended sediment is rapidly ejected and rolled-up around the primary roller, the high sediment concentration is maintained there, as observed in FIG. 5 and FIG. 6. It should be noted that statistically defined turbulent energy may contain effects of large-scale fluctuations of main flows of waves and organized vortices as well as small-scales turbulence directly produced by wave-breaking and seabed boundary layer flows, since the statistical averaging cannot separate turbulence scales. That is, the experimental turbulent energy in FIG. 4, FIG. 5 and FIG. 6 cannot be clearly separated into each contribution of the factors through the wave breaking process. However the large-scale coherence of the observed sediment clouds (FIG. 2 and FIG. 13) are unlikely to be caused by the simple diffusion process due to small-scale turbulence. The analogies of the suspension processes by vortex rings [47] provide the consistent interpretation on the contribution of the vortex-induced flows to suspension process (§3.4). Therefore we conclude that
AC
500
505
28
ACCEPTED MANUSCRIPT
510
the coherent vortex-induced flow is one of the major factors to suspend the sediment. This mechanism to induce sediment suspension in the three-dimensional vortex organization is introduced in a suspension model in the next section.
520
525
In §2, the previous suspension models for wave-breaking turbulence were observed to predict inconsistent sediment concentrations during the experimental breaking process. In §3, the vortex-induced suspension has been identified as a major mechanism excluded in the previous models. In this section, we propose a new approach to parameterize the vortex-induced sediment suspension. This approach analytically describe the vertical velocity induced by streamwise counter-rotating vortices in terms of vortex dynamics. The derived model is introduced into the diffusion-advection equation through parametric relations of the transverse flows and wave breaking conditions (Otsuka and Watanabe [12]).
AN US
515
CR IP T
4. vortex-induced suspension model
4.1. Modification to the advection-diffusion equation of sediment concentration A general form of mass conservation for volume-averaged suspended particles in liquid-solid two-phase flow is written by (14)
M
∂ ρs c + ∇ · (ρs cus ) = 0, ∂t
n
n
n
where us,n and Vn are the velocity and volume of the n-th particle, respectively. Equation (14) for the constant ρs is transformed into
PT
530
ED
where ρs is the density of sediment particle. The concentration is defined as the volume fraction of N particles contained in the reference volume V0 , c = X N N N X X V0−1 Vn , and the mean particle velocity us = (us , vs , ws ) ≡ us,n Vn Vn , ∂ c + ∇ · (c us ) = −∇ · c0 u0s ∂t
(15)
CE
via Reynolds decomposition based on the ensemble average. By introducing a gradient diffusion model, c0 u0s ≈ −s ∇c, to equation (15), we obtain (16)
Assuming that the horizontal gradients in c are negligibly smaller than the vertical gradient and time-averaging equation (16), the convection-diffusion equation for temporal mean concentration hci is approximated by
AC 535
∂ c + ∇ · (c us ) = s ∇2 c ∂t
∂ ∂ 2 hci hcihws i = hs i , ∂η ∂η 2
(17)
where η is the vertical coordinate with the origin at the undisturbed bed (η = z + h, see definitions in FIG. 3 b). If only the terminal settling velocity is 29
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 15: Schematic illustration of the mechanism of sediment suspension by the vortexinduced flows under breaking waves; the divergent and convergent near-bed flows induced by the counter-rotating streamwise vortices above the bed cause bed load sediment transport toward the converging locations where the sediment is carried by upward flows induced between a pair of the counter-rotating vortices. The suspended sediment may be then carried by the primary vortex flow and rolled up around the roller perimeter.
30
ACCEPTED MANUSCRIPT
545
∂ ∂ 2 hci . hci (hwi − wf ) = hs i ∂η ∂η 2
(18)
Equation (18) describes the explicit contribution of the vertical fluid flow to pick up sediment to balance the settling convection and the turbulent diffusion. Assuming the constant turbulent mixing by breaking wave hs i, as explained in §1.3, the exponential solution for equation (18) is thus derived as
AN US
550
CR IP T
540
considered as the sediment particle velocity, hws i ≈ wf , assuming a steady mechanical balance between gravitational particle settling in still water and spherical particle drag, ρCd wf2 = 4aρs g/3, equation (17) becomes equivalent to the conventional convection-diffusion equation (1). In the steady state of particle motion with velocity hws i in fluid flow with velocity hwi, the particle drag (in proportion to the squared relative velocity) balances with the gravity; 2 ρCd (hwi − hws i) = 4aρs g/3, where Cd is the drag coefficient. Therefore, as2 suming steady particle motion, we get (hwi − hws i) = wf2 , leading the relation hws i = hwi − wf . Equation (17) is now rewritten as
hci = c0 e−
wf −hwi η hs i
.
(19)
M
It should be noted that this solution excludes effects of the cross-shore sediment transport by the undertow and the rolled-up flow induced by the horizontal roller (FIG. 13), which may cause uncertainty of the prediction, as discussed again in §4.5. 4.2. Vortex induced transporting velocity
560
Let us consider N point vortices located at (yi , zi ) with circulation Γi on a (y − z) two-dimensional plane (vortex number i=1, 2, . . ., N , see FIG. 16 a) as a model of a streamwise vorticity field on a transverse-vertical cross-section (as shown in FIG. 12). The velocity (V , W ) induced by these vortices at arbitrary locations (y, z) is determined on the basis of the well-known Biot-Savart integral [49]: N X Γi z − zi V (y, z) = − (20) 2 2 2π (y − yi ) + (z − zi ) i=1
AC
CE
PT
ED
555
565
W (y, z) =
N X Γi y − yi 2π (y − yi )2 + (z − zi )2 i=1
(21)
Assuming a pair of counter-rotating vortices with identical absolute circulation, i.e., Γ = Γ1 = −Γ2 and N =2, are horizontally aligned at the identical vertical level z0 = z1 = z2 (see FIG. 16 b), equations (20) and (21) are reduced to ! z − z0 z − z0 Γ − (22) V =− 2 2 2π (y − y1 )2 + (z − z0 )2 (y − y2 ) + (z − z0 )
31
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 16: Schematic representations of (a) streamwise point vortices located in a transversevertical cores-section, as a model of a vorticity field in FIG. 12, (b) a pair of counter-rotating vortices with identical absolute circulation Γ, located at z = z0 ., and (c) upward induced flow flux for a pair of counter-rotating vortices with spacing λ. Wc indicates the induced velocity at the center of the vortex pair.
32
ACCEPTED MANUSCRIPT
Γ W = 2π
y − y1 2
(y − y1 ) + (z − z0 )
2
−
y − y2 2
2
(y − y2 ) + (z − z0 )
!
(23)
Wc =
575
where λ is the distance of the vortices. If these are assumed to be Rankine vortices with vorticity ω, the circulation at the edge of the vortex core is given by Γ = 2πrc2 ω = 2πrc γ [6], which is used as the representative circulation of the point vortices, where rc is the core radius and the tangential velocity γ = rc ω.
4.3. Model calibration Otsuka and Watanabe [12] experimentally measured the transverse flows induced by the counter-rotating vortices by using an ultrasonic velocity profiler, and statistically parameterized the characteristic length (λ), time (τ ), and velocity (γ) of the transverse flows in breaking waves. These characteristic values exhibit in-phase variations; the shortest length and time scales are achieved at the wave front, and they gradually increase to become the longest length and time scales behind the wave crest. If these scale properties are assumed to be maintained near the bed, the distance between the vortices, the duration that the flow is influenced by the vortices, and the tangential velocity of the vortices can be approximated by the dimensionless empirical parameters [12]; the maximum length λ/S ≈ −0.64Fr + 0.61, the maximum √ time τ /T ≈ −0.77Fr + 0.66, and the maximum transverse velocity Fr = γ/ gh ≈ 0.47ξ + 0.09, where T is the wave period, ξ is the surf-similarity parameter (5) and the reference length S = 0.25 (m). Here λ/S, τ /T and Fr are physically interpreted as the relative transverse spacing of the counter-rotating flows to the flume width, the relative duration of the transverse flow to the wave period, and the absolute transverse velocity with respective to the wave celerity, respectively. Otsuka and Watanabe[12] compared the spacing between the counter-rotating flows observed in two wave flumes with different width (S), and found λ is independent on S for the same wave-breaking conditions; that is, the organization of the counter-rotating vortices is inherently determined by the breaker conditions regardless of the flume width. It should be noted that the above empirical parameters are available within the range of 0.20 ≤ ξ ≤ 0.73 in the cases of regular wave breaking over an uniform beach slope. Substituting λ and γ into equation (24), Wc at the bottom, z = −h, is parameterized by
CE
590
PT
585
ED
M
580
(24)
AN US
570
λ 2Γ , 2 π λ + 4 (z − z0 )2
CR IP T
The transverse velocity, V , is canceled out at the center of the point vortices, y = (y1 + y2 ) /2 , while the induced vertical velocity there, Wc , is given by
AC
595
600
Wc ≈ 4rc
p
ghFr
S (−0.64Fr + 0.61) 2
2,
S 2 (−0.64Fr + 0.61) + 4 (−h − z0 )
(25)
where the parameterized Froude number Fr = 0.47ξ + 0.09. The core radius, rc , may depend on a breaker. In this study, it was estimated to be rc ≈ 0.01 33
ACCEPTED MANUSCRIPT
620
625
CR IP T
615
AN US
610
M
605
m based on the computed results. The unknown travel depth of the vortex is assumed as z0 ≈ −H to be consistent with the computed one. Equation (25) now indicates the parameterized characteristic velocity to vertically carry the sediment beneath the organized vortex structures of breaking waves. The parameters rc and z0 may depend on breaker conditions, especially breaking wave height and depth. These empirical parameters may need to be redefined for larger scale experiments or in-situ observations. Now we consider the mean upward carrier flow contributing upward sediment flux from the near-bed region. As already discussed, when the counter-rotating vortices approach the sea bed, the bottom sediment is driven by the near-bed convergent flow for being deposited and ejected upward between the vortex pair, while inner water mass is transported downward between the next vortex pair with opposite vorticity orientation to create divergent near-bed flow sweeping the sediment away (see FIG. 12 and FIG. 15). In an equilibrium state of sediment concentration following the exponential distribution, the near-bed concentration may be much higher than the upper one away from the bed, and thus the upward induced flow gives a major contribution to the net vertical sediment flux. In the simplest case of clear water (with no suspended sediment) above the vortices, the downward induced flow provides no contribution to the vertical sediment transport and thus the net sediment flux is defined only by the upward vertical transport. Assuming that the upward carrier flow flux between the vortex pair Ry is given by y12 W dy ≈ Wc λ (see FIG. 16 c), the temporal mean of the upward induced carrier velocity, hWc i, per unit width containing a pair of the counterrotating vortices (2λ) may be approximately given by Wc Wc τ ≈ (−0.77Fr + 0.66) , 2 T 2
(26)
ED
hWc i ≈ α
PT
CE
630
where α is the empirical parameter representing a ratio of sediment concentrations in a near-bed region and far from the bed; α = 1.0 for the clear water case, and α = 0.0 for the case of complete sediment exchange (no net sediment flux) in the uniform vertical profile of concentration. In this study, α =1.0 is used for simplicity. Identifying hwi with the time-averaged upward fluid flow induced by the counter-rotating vortices at the bed, hWc i, equation (19) is rewritten by hci = αc0 e−
640
.
(27)
Equation (27) provides the mechanical contributions of the vortex-induced flows, computationally supported in §3, to the vertical distribution of suspended sediment, experimentally observed in §2. Following Larson and Kraus [21], equation (8) is used as the mixing coefficient hs i for the turbulence caused by wave breaking in this study. FIG. 17 shows the cross-shore distributions of the predicted vortex-induced velocity for the experimental wave cases. It should be noted that the model assumption is acceptable in the shoreward region from the wave-plunging location (x ≥ 0.6 m) where the counter-rotating vortices begin to be produced after
AC 635
wf −hWc i η hs i
34
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 17: Cross-shore variations of parameterized mean vertical velocity induced by the counter-rotating vortices, hWc i, for the experimental cases.
35
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ED
Figure 18: Comparisons of the attenuations rate predicted by the current model and Larson and Kraus [21]; case 1 (red), case 2 (blue), and case 3 (black).
PT
CE
645
wave plunging. The sediment receives vertical drag in the induced flows with hWc i against the mean settling velocity wf =0.0248 m/s. hWc i slightly slower than wf are maintained in the transition region (0.6 m≤ x <1.2 m), and gradually decrease during the wave-breaking process; that is, the upward drag due to hWc i significantly decelerates the mean settling behaviors of the suspended sediment in the transition region, while this effect is reduced in the bore region. In the spilling wave-breaking case (case 1), the induced velocity is smaller than the plunging-wave cases (cases 2 and 3) over the entire surf zone because the effects of the vortices are smaller in this case, resulting in less suspension (see FIG. 4).
AC
650
4.4. Model validation with experimental data
655
FIG. 18 compares the current attenuation model hac i = − (wf − hWc i) /hs,lk i with alk by Larson and Kraus [21] in addition to the experimental results. Because our model uses a mixing coefficient identical to that used by Larson and Kraus [21], equation (8), the difference between the models explicitly indicates 36
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ED
Figure 19: Comparisons of the predicted attenuation rate predicted by the current and previous models with the experimental results; case 1 (red), case 2 (blue), and case 3 (black).
PT
CE
660
the quantitative contribution of the vortex-induced flows to the sediment suspension. The current model ac for the plunging-wave cases (cases 2 and 3) in the transition region are consistent with the experimental attenuation rates, and thus the prediction is significantly improved over that provided by the method of Larson and Kraus [21] predicting much larger attenuation, which suggests that vortex-induced flow is an important factor for describing the concentration of the suspended sediment in the surf zone. The improvement of the prediction for spilling waves producing weaker vortices only near the free surfaces, which may violate the current model assumptions, however, is inadequate. It should be noted that, as the current model assumes that the turbulent mixing is independent on z (constant turbulent mixing), the attenuation rate is also independent on the reference concentration c0 in our approach. FIG. 19 shows the predicted attenuation rates of the current model compared with the previous models by van Rijn [18], Larson and Kraus [21], and Goda [29]. As explained above, while the current model for plunging-wave cases describes attenuations consistent with experimental values, the predicted atten-
AC
665
670
37
ACCEPTED MANUSCRIPT
690
695
CR IP T
5. Conclusions
PT
700
Limitations of the proposed model and uncertainties of the prediction are discussed in this section. Since we followed the conventional assumption of negligibly small cross-shore variations in hci and solved one-dimensional advectiondiffusion equation for the mean concentration, any effects of cross-shore transport by the undertow and the rolled-up flow around the primary roller were excluded from the prediction. As the travel depth of the vortex, z0 , and the core radius, rc , used for determining the vortex-induced flow (equation (25)), may depend on a scale of breaker, these parameters should be reexamined through a parametric study for arbitrary wave-breaking cases. The parametric length, time and velocity sales of the transverse flows, proposed by Otsuka and Watanabe [12], are limited to use in the range of the experimental surf similarity parameter, 0.20 ≤ ξ ≤ 0.73, for regular wave-breaking over a uniform sloping breach. Statistical uncertainties of the near-bed concentration, observed in FIG. 7, may be relevant to uncertain strength of the breaking-wave induced turbulence and vortices as well as uncertain local location where they arrive on the bed. The other possible parameters which may affect sediment transport in the surf zone, such as mobility of bottom sediment, local deformation of seabed, irregularity of incident waves, are excluded in the model, which also needs to be taken into account for further improvement of the prediction in future study.
AN US
685
4.5. Limitation of model parameters
M
680
ED
675
uations are all much larger in the previous models. Wang [22] also compared the model predictions of vertical profiles of the sediment concentration in the surf zone and found the both models by van Rijn [18] and Larson and Kraus [21] underpredict the concentration, which may be due to the models overestimating the attenuation. The proposed model including the vortex-induced suspension mechanism, supported by experimental and computational evidence in §2 and 3, reasonably predicts the mean suspension concentration under violent breakers.
CE
In this study, physical and computational experiments were performed to determine the mechanical contributions of counter-rotating vortices that are sequentially produced during repeated splash-up cycles to sediment suspension in the surf zone. Experimental measurements of the surface elevation, sediment concentration, and fluid velocity distribution revealed that hight sediment concentration appears at neither the wave front following the shoreward velocity nor the region containing strong turbulence behind the wave crest. Vertical profiles of the observed time-averaged sediment concentrations are well defined by exponential attenuations regardless of the breaker type. Previous models fail to predict the measured attenuation rates, causing underpredictions of the sediment concentration in the surf zone, as also observed by Wang [22] and van Rijn [18]. These experimental results indicate another mechanism inducing suspension in addition to the turbulence caused by seabed boundary layers and breaking waves.
AC
705
710
38
ACCEPTED MANUSCRIPT
730
735
745
PT
ED
740
CR IP T
725
AN US
720
We investigated transitional concentrations of passively transported numerical dye, as the simplest model of dilute particle-laden flows involving very fine sediment, in breaking-wave-induced vortex structures in LES computational experiments. When vortex loops with streamwise counter-rotating vortices are stretched obliquely downward between the adjacent primary rollers and approach the bed, transverse dye transport driven by divergent and convergent near-bed flows are induced by the counter-rotating vortices over the bed, resulting in the formation of regular patterns of transverse depositions at the convergent points. The upward flows induced between the counter-rotating vortices carry the deposited dye and eject it upward. A rolled-up coherent pattern of the dye concentration is formed by the advection around the primary roller perimeter. Analogies with a vortex structure involving a vortex ring, wrapping an azimuthal array of the counter-rotating vortices, and the local suspension induced by the vortices [47] support our interpretation of the vortex-induced suspension under breaking waves. While this experiment omits gravitational settling behaviors of sediment, identical contributions of the counter-rotating vortices to the initial displacement and suspension may result in analogous upward ejections of sediment due to obliquely descending eddies, as observed in FIG. 2 and FIG. 13. In addition to the effects of settling and breaking-wave-induced turbulence, the mechanism to initiate suspension due to vortex-induced flows was incorporated into a suspension model in terms of vorticity dynamics with parametric characteristics of transverse flows in breaking waves. The proposed model reasonably predicts the observed exponential attenuation rates for violent breakers and significantly improves the prediction accuracy. Note that the validity of the proposed model is supported within the conditions of the current experiments and the previous parametric study[12]. The model includes uncertainties associated with cross-shore sediment transport, local erosion of seabed, breaker-dependent vortex cores and travel depth of the vortices.
M
715
Acknowledgments
CE
This research was supported by JSPS Grant-in-Aid for Scientific Research (15H04043).
AC
A. Comparison of computational and experimental turbulences
750
The turbulence statistics in the laboratory experiment is compared with the computed flows in this section. In the physical experiment, Reynolds decomposition, based on the ensemble averaging, was used to define the turbulence energy; k=
1 0 0 u ·u , 2 39
(28)
ACCEPTED MANUSCRIPT
770
775
CR IP T
PT
780
AN US
765
M
760
ED
755
where the velocity fluctuation u0 = u − u, and u is the ensemble-averaged velocity. On the other hand, in LES, the sub-grid turbulence energy, ksgs , is defined by 1 1 00g 00 e·u e) ≡ (29) u ·u , ksgs = (ug ·u−u 2 2 ˜ and u ˜ is the grid scale velocity. where the sub-grid scale velocity u00 = u − u While ksgs is the kinetic energy of small-scale turbulence with the length scale less than the grid width ∆ ≈ 6 mm, the statistical definition k may contain large-scale velocity fluctuations of main flows of waves and large scale vortices since the statistical average cannot separate turbulence scales. Therefore ksgs cannot be directly compared with k. The turbulence statistics based on the ensemble averaging for the LES results cannot be obtained straightforwardly since no variance has been computed in any trials with the the identical initial conditions. Therefore it was difficult to obtain the ensemble-averaged representation of the current LES results for fairly comparing with the experimental one. On the other hand, a phase-averaging operation over waves has been also used for estimating the flow statistics in breaking waves (e.g. Ting and Kirby[50], Cox and Kobayashi [51], Kimmoun and Branger [52]). Although the phase averaging may result in a certain difference in turbulence components from the ensemble one because of the a-periodic nature in breaking point and wave period [53], fundamental features of statistical flow variations estimated from the both definitions may be comparable in the regular breaking process. In order to estimate the statistical representation of our LES results, we performed the phase-averaging operation to the computed flows in the same manner as Watanabe et al.[4]. The instantaneous grid-scale velocity, b e , is redefined in terms of the phase-averaged grid-scale velocity u e by u e = u 000 000 b e +u e , where the superscript indicates the deviation from the phase-averaged u ˜ + u00 , we find u b = quantity. Taking the phase-average of the relation u = u 000 b c00 . As u = u e+u b + u , the turbulent energy based on the phase-averaged kp u is described as
CE
kp =
1 000d 000 1 000d 000 d 1 c00 c00 b c00 e ·u e + ksgs − u · u − u e·u . u ·u = u 2 2 2
(30)
AC
c00 is much smaller than the other terms, the above equation may be Assuming u approximately reduced to
785
kp ≈
1 000d 000 d e ·u e + ksgs . u 2
(31)
√ Following Zhou et al. p [11], the experimental turbulent intensity 2k and the computational one 2kp are defined to compare in FIG. 20, together with b and u. We find overall features of the temporal variations in the the velocity u computed velocity are consistent with the experimental ones, and however the computed velocity tends to overestimate with the wave propagation. The computed turbulent intensity agrees with the experimental one at the early breaking 40
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 20: Time records of computed phase-averaged results (solid line) and experimental ensemble-averaged ones (broken line) at η=4.5 cm; dimensionless horizontal velocity (top), vertical velocity (middle) and turbulent intensity (bottom).
41
ACCEPTED MANUSCRIPT
800
805
CR IP T
795
process (x = 0.4 and 0.6 m). In the later stage, while the overestimations are observed at x = 0.8 and 1.1 m, the both intensities present the rapid increase from the identical phases; that is, the inception of the computed turbulent generation coincides with the experimental one. Since the current model assumes a single-liquid-phase flow, any effects of sediment have been excluded from the computation. On the other hand, the experimental incident wave energy should be spent for lifting sediment up [23]. In addition, the experimental flow field may be modified by multiple effects of suspended sediment, including additional dissipation due to particle drag, generation of additional turbulence, particle settling, and mobility of seabed. These effects may be associated with the overestimations of the computed velocity and turbulent intensity in x ≥ 0.8 m where may occur successive modifications of flows during the breaking process. In order to precisely compute the turbulent flow with sediment throughout the surf zone, appropriate modeling of sediment-turbulence interaction is required. However the computed flow statistics at the early stage of wave breaking presents major features of the experimental turbulence, indicating the validity of the current computation under the assumption of minor effect of sediment at the early breaking process.
AN US
790
References
[1] D.H. Peregrine, Breaking waves on breaches, Annu. Rev. Fluid Mech. 15 (1983) 149–178.
M
810
815
ED
[2] I.A. Svendsen, P.A. Madsen, J.B. Hasen, Wave characteristics in the surf zone, Proc. 16th Coastal Engineering Conf. (1978) 520–539. [3] K. Nadaoka, M. Hino, Y. Koyano, Structure of the turbulent flow field under breaking waves in the surf zone, J. Fluid Mech. 204 (1989) 359–387. doi:10.1017/S0022112089001783.
PT
[4] Y. Watanabe, H. Saeki, G. J. Hosking, Three-dimensional vortex structures under breaking waves, J. Fluid Mech. 545 (2005) 291–328. doi:10.1017/ S0022112005006774. [5] T. Sarpkaya, Vorticity, free surface, and surfactants, Annu. Rev. Fluid Mech. 28 (1996) 83–128. doi:10.1146/annurev.fl.28.010196.000503.
CE 820
AC
[6] T. Sarpkaya, P. Suthon, Interaction of a vortex couple with a free surface, Experiments in Fluids. 11 (1991) 205–217.
825
[7] M. Brocchini, D. H. Peregrine, The dynamics of strong turbulence at free surfaces. part 1. description, J. Fluid Mech. 449 (2001) 225–254. doi: https://doi.org/10.1017/S0022112001006012. [8] Y. Watanabe, N. Mori, Infrared measurements of surface renewal and subsurface vortices in nearshore breaking waves, J. Geophys. Res. Oceans 113 (2008) C07015. doi:10.1029/2006JC003950. 42
ACCEPTED MANUSCRIPT
835
840
[9] A. Saruwatari, Y. Watanabe, D. M. Ingram, Scarifying and fingering surfaces of plunging jets, Coastal Eng. 56 (2009) 1109–1122. doi:http: //dx.doi.org/10.1016/j.coastaleng.2009.08.007. [10] P. D. LeClaire, F. C. Ting, Measurements of suspended sediment transport and turbulent coherent structures induced by breaking waves using two-phase volumetric three-component velocimetry, Coastal Engineering 121 (2017) 56–76. doi:http://dx.doi.org/10.1016/j.coastaleng. 2016.11.008.
CR IP T
830
[11] Z. Zhou, T.-J. Hsu, D. Cox, X. Liu, Large-eddy simulation of wave-breaking induced turbulent coherent structures and suspended sediment transport on a barred beach, J. Geophys. Res. Oceans, 122 (2017) 2017–235 doi: 10.1002/2016JC011884.
AN US
[12] J. Otsuka, Y. Watanabe, Characteristic time, length and velocity scales of transverse flows in the surf zone, Coastal Eng. J. 57 (02) (2015) 1550006. doi:10.1142/S0578563415500060. [13] H. Rouse, Fluid mechanics for hydraulic engineers, Dover, New York, 1961.
[15] L. Amoudry, T.-J. Hsu, P. L.-F. Liu, Schmidt number and near-bed boundary condition effects on a two-phase dilute sediment transport model, Journal of Geophysical Research, 110, C09003 (2005). doi:10.1029/ 2004JC002798.
ED
850
[14] L. C. van Rijn, Sediment transport, Part II: Suspended load transport, Journal of Hydraulic Engineering, 110 (11) (1982) 1613 – 1641.
M
845
[16] H. Lundgren, Turbulent currents in the presence of waves, Coastal Engineering Proceedings 12 (1970) 623–634.
[18] L. C. van Rijn, Principles of sediment transport in rivers estuaries and coastal seas, Aqua publications, 1993.
CE
855
PT
[17] I. G. Jonsson, Measurements in the rutbulent wave bounrary layer, IAHR 10th Congr. 1 (1963) 85–92.
AC
[19] P. Nielsen, Field measurements of time-averaged suspended sediment concentrations under waves, Coastal Engineering 8 (1) (1984) 51–72. doi: http://dx.doi.org/10.1016/0378-3839(84)90022-X.
860
865
[20] P. Nielsen, Three simple models of wave sediment transport, Coastal Engineering 12 (1) (1988) 43–62. doi:http://dx.doi.org/10.1016/ 0378-3839(88)90014-2.
[21] M. Larson, N. Kraus, Estimation of suspended sediment trapping ratio for channel infilling and bypassing, US Army Corps of Engineers, Technical Notes ERDC/CHL CHETN-IV-34 (2001) 1–9.
43
ACCEPTED MANUSCRIPT
[22] P. Wang, W. Yuan, L. Min, Measuring and modeling suspended sediment concentration profiles in the surf zone, Journal of Palaeogeography 1 (2) (2012) 172–192. doi:http://dx.doi.org/10.3724/SP.J.1261. 2012.00013.
875
[23] Y. Goda, Empirical formulation of sediment pickup rate in terms of wave energy flux dissipation rate, Coastal Engineering Journal 55 (04) (2013) 1350012. doi:10.1142/S0578563413500125.
CR IP T
870
[24] H. Gotoh, Accurate particle method for computational sediment dynamics, Proc. two-phase modelling for sediment dynamics in geophysical flows THESIS-2016 (2016) 3 –6.
880
AN US
[25] T.-J. Hus, Z. Cheng, J. Chauchat, X. Yu, A multi-dimensional multiphase sediment transport modeling framework - an open-source community modeling effort, Proc. two-phase modelling for sediment dynamics in geophysical flows THESIS-2016 (2016) 7 –10. [26] J. Fredsøe, R. Deigaard, Mechanics of Coastal Sediment Transport, Vol. 3 of Advanced Series on Ocean Engineering, World Scientific, 1992.
[27] J. J. Bosman, H. J. Steetzel, Time and bed averaged concentrations under waves (1986) 986–1000.doi:10.1061/9780872626003.074. [28] L. C. van Rijn, A. Kroon, Sediment transport by currents and waves, Coastal Engineering Proceedings (1993) 2613–2628.doi:10.1061/ 9780872629332.199.
M
885
[30] J.A. Battjes, Surf Similarity , Proc. 14th Int. Conf. on Coastal Engineering (1974) 466–480.
PT
890
ED
[29] Y. Goda, Re-evaluation of sediment pickup coefficient in the surf zone based on field measurement and large-scale experiment data , ECOH/YG technical report No.11 (2010), in Japanese.
CE
[31] C.E. Willert, M. Gharib, Digital particle image velocimetry , Experiments in Fluids 10 (1991) 181–193. [32] P. Lubin, S. Glockner, Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments, J. Fluid Mech. 767 (2015) 364–393. doi:https://doi.org/10.1017/jfm. 2015.62.
AC
895
900
[33] Y. Watanabe, A. Saruwatari, D. M. Ingram, Free-surface flows under impacting droplets, J. Comp. Phys. 227 (2008) 2344–2365. doi:http: //dx.doi.org/10.1016/j.jcp.2007.10.020. [34] M.S. Longuet-Higgins, Capillary rollers and bores, J. Fluid Mech. 240 (1992) 659-679.
44
ACCEPTED MANUSCRIPT
[35] V. Yakhot, S. A. Orszag, Renormalization group analysis of turbulence. I. basic theory, J. Sci. Comput. 1 (1) (1986) 3–51.
910
[36] S. Osher, R. P. Fedkiw, Level set methods: An overview and some recent results, J. Comp. Phys. 169 (2001) 463–502. doi:http://dx.doi.org/10. 1006/jcph.2000.6636.
CR IP T
905
[37] T. Yabe, T. Aoki, A universal solver for hyperbolic equations by cubicpolynomial interpolation I. one-dimensional solver, Comput. Phys. Commun. 66 (2-3) (1991) 219–232. doi:10.1007/BF01413793. [38] D. Lakehal, P. Liovic, Turbulence structure and interaction with steep breaking waves, J. Fluid Mech. 674 (2011) 522–577.
AN US
915
[39] P.D. Hieu, K. Tanimoto,, V. T. Ca, Numerical simulation of breaking waves using a two-phase flow model, Applied Mathematical Modeling 28 (2004) 983–1005. [40] T. Suzuki, A. Okayasu,, T. Shibayama, A numerical study of intermittent sediment concentration under breaking waves in the surf zone, Coastal Engineering 54 (2007) 433–444.
920
[41] S. Elghobashi, On predicting particle-laden turbulent flows, Applied Scientific Research 52 (1994) 309–329.
M
[42] C. T. Crowe, T. R. Troutt, J.N. Chung, Numerical models for two-phase turbulent flows, Annu. Rev. Fluid Mech. 28 (1996) 11–43.
[45] D. Liang, L. Cheng, F. Li, Numerical modeling of flow and scour below a pipeline in currents Part II. Scour simulation, Coastal Engineering 52 (2005) 43–62.
CE
930
[44] S. K. Jha, F.A. Bombardelli, Two-phase modeling of turbulence in dilute sediment-laden, open-channel flows, Environ. Fluid. Mech. 9 (2009) 237– 266.
PT
925
ED
[43] J. C. Winterwerp, Stratification effects by cohesive and noncohesive sediment, J. Geophys. Res. 106 (19) (2001) 22559–22574.
[46] J. Jeong, F. Hussain, On the identification of a vortex, Journal of Fluid Mechanics 285 (1995) 69–94. doi:10.1017/S0022112095000462.
AC
[47] R.J. Munro, N. Bethke, S.B. Dalziel, Sediment resuspension and erosion by vortex rings, Physics of Fluids 21 (2009), doi:10.1063/1.3083318 doi: 10.1063/1.3083318.
935
[48] E. Sheila, E. Windnall, J.P. Sullivan, On the stability of vortex rings, Proc. R. Soc. Lond. A. 332 (1973) 335–353. [49] S.E. Widnall, The structure and dynamics of vortex filaments, Annu. Rev. Fluid Mech 7 (1975) 141–165. 45
ACCEPTED MANUSCRIPT
940
[50] F.C. Ting, J.T. Kirby, Dynamics of surf-zone turbulence in a spilling breaker, Coastal Engineering 27 (1996) 131–160.
945
[52] O. Kimmoun, H. Branger, A particle images velocimetry investigation on laboratory surf-zone breaking waves over a sloping beach, J. Fluid Mech. 588 (2007) 353–397. [53] K.-A. Chang, P. L.-F. Liu, Experimental investigation of turbulence generated by breaking waves in water of intermediate depth, Phys. of Fluid 11 (1999) doi:10.106063/1.870198.
AC
CE
PT
ED
M
AN US
950
CR IP T
[51] D.T. Cox, N. Kobayashi, Identification of intense, intermittent coherent motions under shoaling and breaking waves, J. Geophys. Res. 105 C6 (2000) 14223–14236.
46
Vortex-induced suspension of sediment in the surf zone Junichi Otsuka, Ayumi Saruwatari, Yasunori Watanabe PII: DOI: Reference:
S0309-1708(17)30268-3 10.1016/j.advwatres.2017.08.021 ADWR 2948
To appear in:
Advances in Water Resources
Received date: Revised date: Accepted date:
16 March 2017 28 July 2017 2 August 2017
Please cite this article as: Junichi Otsuka, Ayumi Saruwatari, Yasunori Watanabe, Vortexinduced suspension of sediment in the surf zone, Advances in Water Resources (2017), doi: 10.1016/j.advwatres.2017.08.021
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights • A major mechanism of sediment suspension by organized vortices produced under violent breaking waves in the surf zone was identified.
CR IP T
• Effect of the vortex-induced flows was incorporated into a suspension model.
AC
CE
PT
ED
M
AN US
• The model proposed reasonably predicts the measured sediment concentration due to violent plunging waves and significantly improves the underprediction of the concentration produced by previous models.
1
ACCEPTED MANUSCRIPT
Vortex-induced suspension of sediment in the surf zone Junichi Otsukaa , Ayumi Saruwatarib , Yasunori Watanabeb,∗ a Civil
CR IP T
Engineering Research Institute for Cold Region, Hiragishi 1-3-1-34, Sapporo 062 8602, Japan b Hokkaido University, Graduate School of Engineering, North 13 West 8, Sapporo 060 8628, Japan
Abstract
ED
M
AN US
A major mechanism of sediment suspension by organized vortices produced under violent breaking waves in the surf zone was identified through physical and computational experiments. Counter-rotating flows within obliquely descending eddies produced between adjacent primary roller vortices induce transverse convergent near-bed flows, driving bed load transport to form regular patterns of transverse depositions. The deposited sediment is then rapidly ejected by upward carrier flows induced between the vortices. This mechanism of vortexinduced suspension is supported by experimental evidence that coherent sediment clouds are ejected where the obliquely descending eddies reach the sea bed after the breaking wave front has passed. In addition to the effects of settling and turbulent diffusion caused by breaking waves, the effect of the vortexinduced flows was incorporated into a suspension model on the basis of vorticity dynamics and parametric characteristics of transverse flows in breaking waves. The model proposed here reasonably predicts an exponential attenuation of the measured sediment concentration due to violent plunging waves and significantly improves the underprediction of the concentration produced by previous models.
PT
Key words: sediment suspension, counter-rotating vortices, breaking waves 2016 MSC: 00-01, 99-00
CE
1. Introduction
AC
In the surf zone, a shoaling wave crest projects forward as an overturning jet and splashes onto the forward water surface [1]. The rotational fluid motion within the jet surrounding the air tube governs the flow involving a primary roller vortex at an early stage of wave breaking (see FIG. 1). Secondary jets are sequentially ejected to produce another roller vortex in the repeating splashing process, resulting in an array of horizontal roller vortices in the transition
5
∗ Corresponding
author
Preprint submitted to Advances in Water Resources
September 21, 2017
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 1: Schematic illustration of evolution of breaking waves; plunging breaker (top) and spilling breaker (bottom).
15
M
ED
10
region. The energetic splashing waves then culminate in turbulent bores propagating toward the shore in a bore region [2]. Nadaoka et al. [3] observed the formation of three-dimensional vortex structures involving obliquely descending eddies (ODEs) that developed along the principal axis of fluid strain, behind the rollers. They suggested that the ODEs contribute to the sediment suspension observed in the surf zone. This study focuses on mechanical roles of the breaking-wave-induced vortices to the sediment suspension and prediction of the sediment concentration in the surf zone.
FIG. 2 shows typical sequential images of experimental breaking waves at the plunging location over the sand bed (see also supplementary movie 1). After the wave front passes, FIG. 2(a), sediment begins to be picked up on the bed at the rear end of the aerated area, where the ODEs may entrap air bubbles, in FIG. 2 (b). The sediment is then rapidly transported upward with a rotating motion (FIG. 2 (c), and see also supplementary movie 1) and ejected up to the surface at (d). Watanabe et al. [4] identified the mechanism that forms the ODEs during the wave-breaking process; that is, initial primary vorticity changes in orientation in a stagnation-point flow induced between the adjacent roller vortices, and the re-oriented vortex filaments are stretched in the direction of the maximum eigenvector of fluid strain, resulting in a transverse array of counter-rotating vortices (that are identical to ODEs). When counter-rotating vortices approach a free-surface, the surface above the vortices is entrained into the inner water mass and is thus depressed, forming so-called scars (Sarplaya and Suthon[6],
AC
CE
20
PT
1.1. Obliquely Descending Eddies
25
30
3
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 2: Sequential images of experimental breaking waves passing over a sand layer in a wave flume shown in FIG. 3 (time interval of 0.2 s); (a) wave plunging results in the formation of secondary splash-up jets and entraps a rotating air tube in the inner water mass. (b) The ODEs, containing air bubbles, reach the bed and begin to pickup local sediments after passing the wave front (see arrow). (c, d) The suspended sediments are transported upward with a rotating motion. See also supplementary movie 1.
4
ACCEPTED MANUSCRIPT
45
CR IP T
40
AN US
35
Sarpkaya[5], Brocchini and Paregrine[7]). In the case of breaking waves, the subsurface counter-rotating vortices induce convergent flows to entrain the surface to bulk water and create the scars behind the wave crest (Watanabe and Mori [8]), and the development of scars on the jet surfaces penetrates the jet, fragmenting them into finger jets (Saruwatari et al. [9]). In addition to these contributions of breaking-wave-induced vortices on the surface boundary, several recent studies demonstrated explicit roles of the counter-rotating vortices (or ODEs) in the dynamics of sediment-laden flows near the bottom boundary. LeClaire and Ting [10] found high correlations between the population of suspended sediment particles and vorticity when the sediment was observed to be trapped in counter-rotating vortices. Zhou and Hsu [11] computationally identified that upward and downward vertical velocities caused by the penetration of ODEs in a near-bed region enhance sediment suspension next to the edge of the ODEs. According to Otsuka and Watanabe [12], regular patterns of divergent and convergent transverse flows appear in an array of counter-rotating vortices near the bed under breaking waves, suggesting that corresponding upward and downward vertical flows are also induced between the vortices. 1.2. Rouse distribution of sediment concentration
A general description of the mean concentration of suspended sediment, hci, has been provided by Rouse [13]. In an equilibrium state that the rate of settling equals the rate at which the sediment is lifted by turbulent diffusion,
M
50
wf hci = −hs i
(1)
ED
where η is the vertical coordinate with the origin at the undisturbed bed and wf is the settling velocity. h i indicates the temporal mean quantity. The sediment diffusion, hs i, is generally related to the turbulent diffusion of fluid, hf i, in terms of Schmidt number Sc as hs i = hf i/Sc . In steady open√channel flow, 1 τ0 assuming the logarithmic regime of the near bed flow du dη = κ η , the wellknown Rouse distribution [13] is given as the solution of equation (1), assuming Sc = 1; R η0 (h − η) hci = , (2) c0 η (h − η0 )
CE
PT
55
dhci , dη
where c0 is the reference concentration at the reference level η0 , h is the water p depth, R = wf /κ τ0 /ρ is the so-called Rouse parameter, κ is the von K´ arm´ an’s constant, and τ0 is the wall shear stress. In this case, the diffusion of fluid momentum takes a form of parabolic distribution; η κu0 , (3) hf i = η 1 − d p where u0 = τ0 /ρ is the friction velocity. In general, since Sc depends on c/c0 , hs i is also described as a function of the relative concentration c/c0 [14, 15].
AC
60
65
5
ACCEPTED MANUSCRIPT
1.3. Sediment concentration in the surf zone
hci = c0 e−wf η/hs i ,
80
CE
95
PT
90
ED
M
85
AC
100
105
(4)
which is the solution of equation (1) assuming the constant hs i. The identical profile of the concentration has been observed in the surf zone [19, 22], indicating the constant near-bed sediment mixing reasonably defines the breakingwave-induced turbulence, rather than the parabolic mixing (3) associated with logarithmic turbulent boundary layer. In model analyses of the suspended coastal sediment, the sediment mixing has also been studied for non-breaking waves (Lundgren [16], Jonsson [17], van Rijn [18]), and for vortex ripples under waves (Nielsen [19, 20]). Van Rijn [18] proposed a three-layer model of wave-induced sediment concentration. Larson and Kraus [21] modeled the vertical attenuation rate for an exponential concentration profile of the suspended sediment in the surf zone. Wang [22] compared the vertical profiles of the sediment concentration predicted by these models with experimental results in the surf zone, and found that both predictions were significantly lower than the measured results, regardless of breakers. Goda [23] integrated the previous datasets of the sediment concentration observed in the surf zones, leading to an empirical formula for sediment pickup rates. The remarkable variabilities in the observed concentration from the formula (0.18 – 4.2 times disparity), reported by Goda [23], suggest that uncertainties in predicting the suspension processes in complex surf zone flows remain. All these previous models exclude explicit effects associated with breaking-wave-induced vortices to carry sediment and maintain suspension against gravity, observed in FIG. 2, which may be one of the major factors that needs to be identified to improve the prediction of the sediment concentration in breaking waves. A recent progress of computational multiphase flow modeling, especially in a Lagrangian particle method[24] and turbulence-sediment coupled approach[25], was discussed at THESIS-2016 (the 3rd symposium on two-phase modeling for sediment dynamics in geophysical flows). While new numerical models are expected for further understanding of the sediment transport in the surf zone, the current study focuses on semi-analytical improvement of the previous empirical models in terms of vorticity dynamics in the surf zone. In this paper, fundamental statistical features of the suspended sediment in turbulent flows under breaking waves are experimentally investigated in §2. The mechanical effects of the counter-rotating vortices produced during the wavebreaking process to sediment transport are discussed through computational experiments using a Large Eddy Simulation (LES) of breaking waves in §3. In §4, a novel model of the vortex-induced vertical flow driving suspension was then developed based on the experimental and computational findings. The
AN US
75
CR IP T
70
Since unsteady, oscillatory boundary layer flows are produced in shallow water wave field, the velocity profile of the steady turbulent boundary layer flow, used in §1.2, is no longer acceptable. Bosman and Steetzel[27] found that the experimental mean concentration under coastal waves significantly deviates from the Rouse distribution (2) , and it follows the exponential distribution;
6
ACCEPTED MANUSCRIPT
110
reliability of the proposed model is discussed through comparisons with the experimental results as well as with the predictions of the previous models. Our findings and conclusions are summarized in §5.
CR IP T
2. Laboratory Experiment 2.1. Experimental setup
130
135
AN US
AC
CE
140
M
125
ED
120
The experiments were performed in a 24-m long wave flume, 0.4 m wide and 1.0 m deep, fitted with a piston type wave maker and active wave absorption (see FIG. 3 a). A sand layer was installed in a rectangular sand pit (3 m long, 0.4 m wide, and 0.1 m deep) to cover the surf zone of the 1/20 slope beach. The median diameter, d50 , of the sand was 0.20 mm. We defined the coordinate system for the experiment with the origin at the point of wave-breaking in the horizontal shoreward x axis and at the still water level in the vertical z axis (see FIG. 3 b). The surface elevation ζ(x, t) was defined as the vertical displacement of the free surface about the still water level, z = 0. The sediment concentration, fluid velocity, and surface elevation were measured at the same cross-shore locations that were traversed at 100-mm intervals from the breaking point (x = 0) in the x direction over the surf zone. The concentration of suspended sediment was measured using an optical concentration sensor (PMT5-50, Kenek) that was vertically traversed at 10-mm intervals from the bottom to the level of the wave trough. Particle Imaging Velocimetry (PIV) estimates planar distributions of the fluid velocity from sequential images of tracers recorded at the measurement sites (see FIG. 3 b and c). A continuous wave YAG laser sheet (DPGL-8W, Japan Laser), which was guided into the water through an optical fiber, was emitted shoreward from the offshore side of the breaking point to illuminate the tracer particles in a horizontal-vertical cross-section for recording by an 8-bit high-speed video camera (FASTCAMSA3, Photron) (see FIG. 3 b and c). The field-of-view of the camera was 120 mm× 60 mm. Neutral buoyant fluorescent particles approximately 20 µm in diameter were used as tracers of the fluid flow. Laser light (with a wavelength of 532 nm) was reflected by the suspended sand, entrained air bubbles, and the free-surface of the waves. A camera equipped with a high-pass optical filter (cut-off wavelength < 580 nm) prevented the recording of reflected light, and captured only the laser-induced fluorescent light (with a light spectrum peak of 650 nm) emitted from the fluorescent tracers. Eight-bit, 1024×512 pixel images were recorded at 250 Hz with an exposure time of 1/1500 s and stored on a PC connected to the camera as uncompressed bitmaps. A capacitance-type wave gage was used to record the simultaneous surface elevation at the measurement sites. Analog signals transmitted from the wave gages and concentration sensor were converted to digital data with a sampling frequency of 100 Hz and saved on a PC. When the first wave passed the wave gage installed offshore (see FIG. 3 a), a trigger signal was sent to the high-speed video camera and PC to simultaneously start recording the images and surface elevation data. This synchronized the wave phase of the 30 trials performed for each experimental case to allow
PT
115
145
150
7
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 3: Experimental setup; (a) wave flume, (b) cross-shore locations of measurements and definitions of the coordinate system and surface elevation ζ, (c) photograph of the measurement site (high-speed camera records red fluorescent light emission from tracers on a green laser sheet).
8
ACCEPTED MANUSCRIPT
Table 1: Wave conditions
T (s)
1 2 3
1.4 1.8 2.0
*1
Hb (cm)*2
hb (cm)
12.5 14.7 13.7
13.8 15.0 14.1
*3
*4
bed slope
ξ
1/20 1/20 1/20
0.247 0.293 0.337
*1
Wave period Breaking-wave height *3 Breaking-water depth *4 Surf similarity parameter *2
breaker type Spilling Plunging Plunging
CR IP T
Case
Table 2: Characteristic location and depth of breaking waves in the surf zone (see FIG. 1).
3
Bore Region
(xp ≤ x < xB )
(xB ≤ x)
0.0
–
13.8
–
0.0
60.0
15.0
12.0
0.0 14.1
0.0 – 122.0
122.0 – 276.0
13.8–7.7
7.7 –0.0
0.0 – 110.0
110.0 – 300.0
15.0–9.5
9.5 –0.0
68.0
0.0 – 124.0
124.0 – 282.0
10.7
14.1–7.9
7.9 –0.0
PT
for statistical analysis of the results on the basis of ensemble averaging. The sand bed surface was flattened before each trial to achieve identical initial bed conditions for every trial. We performed three experimental cases of wave breaking (see Table 1). The wave-breaking process depends on the breaker type; plunging jets are sequentially splashed to produce multiple rollers in a plunging breaker, as illustrated in FIG. 1, while a smaller roller vortex produced at the wave crest spilled down on the wave face in a spilling breaker. Table 2 summarizes the experimental locations characterizing the surf zone flows; the wave plunging point (xp ), transition region (xp ≤ x ≤ xB ) and the bore region (xB ≤ x) (see also FIG. 1). The breaker type is commonly characterized by a surf similarity parameter [30] defined by p (5) ξ = tan θ/ Hb /L0 ,
CE
155
Transition Region
(xp )
AN US
2
x (cm) h (cm) x (cm) h (cm) x (cm) h (cm)
Plunging Point
M
1
Breaking Point
ED
Case
AC
160
165
where θ is the bottom slope angle and L0 is the offshore wavelength. The breaker type and ξ for each case are summarized in Table 1.
9
ACCEPTED MANUSCRIPT
180
CR IP T
175
AN US
170
The acquired images were calibrated with real measurements at a 0.1-mm/pixel resolution. The image noise was reduced using a median digital filter. In the PIV analysis, a planar distribution of the instantaneous fluid velocity was estimated on the basis of a standard cross-correlation method [31] for 31 × 31 pixels of interrogation windows on two sequential images. The ensemble average was taken over 30 trials to definethe ensemble averaged velocity u and the turbu1 lent energy k = 2 u02 + w02 , where the velocity fluctuation u0 = (u0 , w0 ) was defined as the deviation from u through Reynolds decomposition. We found that the laser light transmission was interrupted in highly aerated areas and thus significant errors occurred there. The unreliable results contaminated by the aeration were eliminated from the statistics when the maximum correlation between particle image windows is less than 0.65, and/or |u − um | > 1.5σ, where um and σ are the mean velocity and standard deviation over the neighboring window locations, respectively. When available trials were less than 30, the measured data was not used for the analysis to ensure statistical reliability of the current results. 2.2. Experimental results
M
185
FIG. 4 shows the ensemble-averaged surface elevation, ζ, fluid velocity, u = (u, w), turbulent energy, k, sediment concentration, c, and the cross-shore distribution of u, k and c averaged over a FOV, for the spilling wave case, when the maximum sediment concentration is achieved. Readers can also see the sequential results throughout the breaking process in supplementary movie 2. While the maximum shoreward velocity appeared behind the wave crest in x ∼ 100 cm, higher turbulent energy extended over a wide region under the wave, 60 ≤ x ≤ 120 cm. The sediment concentration became maximal at a location more distant from the wave crest, x ∼ 40 cm (see FIG. 4 d), indicating no direct correlation of the concentration with the spatial development of shoreward flow and turbulence. In plunging wave cases (see FIG. 5 and FIG. 6), the turbulent energy was locally intensified through repeating splash-up processes (see FIG. 1), transported downward, and widely distributed behind the wave front (see also supplementary movies 3 and 4). We found that a moderate level of turbulent energy (20 – 50 % of the observed maximum turbulent energy) was transported to the bottom, which may affect the diffusion of the suspended sediment. While the sediment concentration extended over a wide region (0 ≤ x ≤ 90 cm for case 2, and 60 ≤ x ≤ 100 cm for case 3), this region does not coincide with where the energetic turbulence appeared (40 ≤ x ≤ 110 cm for case 2, and 60 ≤ x ≤ 180 cm for case 3) nor where shoreward fluid velocity occurred under the wave crest (90 ≤ x ≤ 120 cm for case 2, and 110 ≤ x ≤ 160 cm for case 3). The high concentration was locally maintained over the depth in the duration of t/T ∼ 0.3 − 0.4 after passing the upstream end of the region containing energetic turbulence in both cases (see supplementary movies 3 and 4). These results suggest another mechanism to uplift the suspended sediment and increase the concentration at higher levels in addition to effects of
CE
195
PT
ED
190
AC
200
205
10
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 4: Ensemble-averaged surface elevation (a), velocity vector and contour of turbulent energy (b), sediment concentration (c), and cross-shore distributions of k, c and u averaged over a FOV (d) at the phase in which the maximum concentration is achieved, in case 1 (spilling breaker). See also supplementary movie 2.
11
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 5: Ensemble-averaged surface elevation (a), velocity vector and contour of turbulent energy (b), sediment concentration (c), and cross-shore distributions of k, c and u averaged over a FOV (d) at the phase in which the maximum concentration is achieved, in case 2 (plunging breaker). See also supplementary movie 3.
12
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 6: Ensemble-averaged surface elevation (a), velocity vector and contour of turbulent energy (b), sediment concentration (c), and cross-shore distributions of k, c and u averaged over a FOV (d) at the phase in which the maximum concentration is achieved, in case 3 (plunging breaker). See also supplementary movie 4.
13
ACCEPTED MANUSCRIPT
Table 3: Experimental data of the reference concentrations and attenuation rates.
case 2
100
120
140
-7.34 -8.55 -5.38 897.5 1147.6 646.7 -10.10 -10.68 -9.20 1631.4 1968.5 1289.7 -4.40 -5.14 -3.28 1128.3 1404.3 853.5
-11.01 -12.17 -8.81 867.0 1162.1 577.1 -9.50 -10.73 -7.73 1614.1 1995.5 1235.3 -9.31 -9.79 -8.52 1532.1 1956.8 1107.8
-13.01 -14.71 -9.25 739.5 1019.3 452.2 -8.35 -7.92 -9.08 1273.7 1557.5 989.8 -12.52 -11.56 -14.80 2029.1 2811.8 1239.9
-11.10 -13.58 -4.63 572.2 815.8 312.3 -9.52 -7.33 -13.98 881.6 1098.2 672.5 -18.91 -19.68 -17.15 1937.5 2727.3 1143.8
-25.57 -25.22 -26.23 657.0 851.1 463.0 -12.00 -14.40 -7.56 622.3 858.2 387.8 -16.37 -17.64 -14.13 1386.8 1899.3 877.7
-22.70 -26.43 -15.53 458.8 657.5 265.3 -13.94 -15.98 -10.29 555.0 756.0 356.0 -26.62 -30.76 -18.42 1328.2 1948.0 728.7
the turbulence caused by wave breaking and the friction in the wave boundary layer. The vertical profile of temporal mean sediment concentration, hci, was conventionally estimated by solving the one-dimensional convection-diffusion equation derived from the continuity equation for sediment, equation (1). As already explained in §1.3, the well-known Rouse distribution of hci for steady turbulent boundary layer flow, equation (2), provide an inaccurate approximation of the one observed in wave field where the sediment is periodically driven by unsteady, oscillatory boundary layer flows over the seabed [27]. In the surf zone where the breaking-wave-induced turbulence and vortices govern the near-bed sediment mixing, the concentration has been observed to follow the exponential distribution (4) (e.g. Nielsen [19], Wang et al. [22]). A number of models for the reference concentration, c0 , and attenuation rate, hai, for the general form of the exponential distribution,
CE
PT
215
80
ED
210
60
M
case 3
hai (m ) hai+ (m−1 ) hai− (m−1 ) c0 (ppm) c0+ (ppm) c0− (ppm) hai (m−1 ) hai+ (m−1 ) hai− (m−1 ) c0 (ppm) c0+ (ppm) c0− (ppm) a (m−1 ) hai+ (m−1 ) hai− (m−1 ) c0 (ppm) c0+ (ppm) c0− (ppm)
40
CR IP T
case 1
−1
AN US
x (cm)
AC
220
225
hci = c0 ehaiη ,
(6)
have been proposed for non-breaking water waves (Nielsen [20], van Rijn [18]) and breaking waves (Larson and Kraus [21]). According to van Rijn [18], the sediment concentration has different vertical profiles in a near-bed mixing layer, governed by turbulence and vortices, and above the layer with orbital fluid motion (see also Bosman and Steetzel [27]). FIG. 7 shows the vertical profiles of the measured time-averaged concentra14
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 7: Cross-shore variations in vertical profiles of the time-averaged sediment concentration (circle) with error bars and the exponential approximation curve (red line); case 1 (left), case 2 (middle), and case 3 (right). The dotted and broken lines represent the exponential approximation curves of the standard deviations above and bellow the mean concentration, respectively.
15
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ED
Figure 8: Cross-shore variations in the predicted attenuation rate by Larson and Kraus [21], Goda [23] and van Rijn [18]; case 1 (red), case 2 (blue), and case 3 (black).
PT
tions in the surf zone, together with the exponential curves, hai, approximated by the least squares method . The error bar represents the standard deviation of the mean concentration. The exponential approximation appropriately describes the vertical distribution of the mean concentration over the measured depth (below the wave trough level) . Van Rijn [18] suggested that the mixing effects in the surf zone can be represented by increasing the mixing layer thickness where the concentration is defined by the exponential distribution, which may support the current results of the exponential attenuation over the depth. We find the standard deviations of the mean concentration appear to be maximal in the near-bed region and decays away from the bed for any cases, suggesting higher uncertainty of the near-bed concentration, which is discussed in §4.5. The attenuation rates for the concentrations at the standard deviations above and bellow the mean concentration, hai+ and hai− , are also well approximated by the exponential curves. All estimated values of the attenuation rate, hai, the reference concentration, c0 , and those estimated from the deviation concentrations (hai+ , hai− , c0+ and c0− ) are summarized in table 3.
CE
230
AC
235
240
16
ACCEPTED MANUSCRIPT
Van Rijn [28] proposed the three-layer model of the sediment concentration. In the near-bed layer where the exponential form is assumed, the sediment mixing coefficient is modeled as hs,vr i = 0.004D∗ Uδ δs ,
250
where the particle parameter D∗ = d50 (s − 1) g/ν , Uδ is the peak orbital velocity, s is the specific gravity, ν is the dynamic viscosity, the thickness of the 0.5 sediment mixing layer δs = 0.3h (H/h) , h is the water depth, and H is the wave height. Van Rijn [18] suggested 0.05 m ≤ δs ≤ 0.2 m for breaking waves. Larson and Kraus [21] suggested the mixing coefficient hs,lk i = kd h (D/ρ)
1/3
(8)
where the energy dissipation rate D ≈ ∂F/∂x, F is the energy flux, and the dimensionless empirical coefficient kd ≈ 0.03. Goda [23] empirically integrated the observation datasets of the mean sediment concentration, and an empirical relation of the estimated attenuation rates was suggested in Goda [29]: A hag i = − D−0.2 , h
(9)
where A=6. The predicted attenuation rates for van Rijn [28], havr i = −wf /hs,vr i, for Larson and Kraus [21], halk i = −wf /hs,lk i, and for Goda [23], hag i, are shown in FIG. 8. All of these predictions exhibit much higher attenuations than the current experimental attenuations, hai, hai+ and hai− , throughout the surf zone, which may result in an underprediction of the concentration over the depth. Wang [22] came to the identical conclusions that both the predicted concentrations by van Rijn [18] and Larson and Kraus [21] are much lower than the measured concentrations. Van Rijn [18] also suggested a considerable underprediction of the concentration observed in a large-scale wave flume, while the model results were consistent with the measurements made in a small-scale wave flume. In the next section, the possible mechanical causes of the inconsistent predictions and the required parameters to improve the prediction are discussed through computational experiments of material transport in breaking-waveinduced turbulent flows.
CE
270
PT
265
ED
M
260
,
AN US
255
(7) 2 1/3
CR IP T
245
AC
3. Computations
275
280
Three-dimensional vortices are produced during the splash-up wave breaking process (Watanabe et al. [4], Saruwatari et al. [9], Lubin and Glockner [32]), and may enhance sediment suspension (Nadaoka et al. [3], LeClaire and Ting [10], Zhou and Hsu [11]). As the instantaneous structure of vortices causing local suspension is difficult to be experimentally quantified, we perform computational experiments with breaking waves to find the mechanisms of the near-bed material transport induced by the three-dimensional vortices. 17
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
PT
Figure 9: Computational domain.
3.1. Computational procedure
CE
3.1.1. Computing methods LES was performed to compute incompressible turbulent flows under breaking waves in three-dimensional space. The computational scheme is summarized in this section and the full details are described in Watanabe et al. [4, 33]. The filtered Navier-Stokes equation was used as the governing equation;
AC
285
290
˜ Du Dt
= −∇˜ p − ∇ · τ + ∇ · τ˜0 + g,
(10)
˜ represents the velocity, where the variables with a tilde denote filtered values, u p˜ is the pressure, τ is the sub-grid scale stress tensor, τ˜0 is the viscous stress ˜ · ∇ is the filtered material tensor, and g is the gravity vector. D/Dt = ∂/∂t + u derivative. The sub-grid scale viscosity model by Yakhot & Orszag [35] was
18
ACCEPTED MANUSCRIPT
Dφ Dt
305
(11)
A sharp interface model to fulfill normal and tangential dynamic conditions at the surface (Watanabe et al. [33]) was introduced in this study. In this model, the normal jump condition, which defines the pressure at the surface to balance with normal viscous stress and surface tension, was used to compute a Poisson equation for pressure. Since the tangential dynamic condition, imposing zero tangential shear at the free-surface, ensures generation of vorticity on a curved surface [34], the model provides reasonable estimates of the primary spanwise vorticity on highly curved inner surface of the overturning jet, and the following formation of finger jets via surface-vorticity interactions [9]. In the framework of a fractional step method, a discretized form of equation (10) was decomposed into discretized advection and non-advection equations. The Cubic Interpolation Polynomials method (Yabe and Aoki [37]) was used to solve the advection equation, while a predictor-corrector method was used to update the non-advection equation. A Poisson equation for pressure was iteratively solved using a multi-grid method. The advection equation for φ, equation (11), was also computed by the Cubic Interpolation Polynomials method. The current numerical technique was validated in Watanabe et al. [33], and the LES model for breaking-wave-induced turbulence was also validated in Watanabe et al. [4].
CE
320
3.1.2. Computational conditions LES was performed in a Cartesian staggered grid system, 1050×66×74 with a dimensional resolution of 6.0 mm per cell, in a rectangle domain tilted with an angle of 1/20 with respect to the horizontal axis x (see FIG. 9). The grid spacings in wall unit were (∆x∗ , ∆y ∗ , ∆z ∗ ) = (60.1, 61.5, 61.4), and an ODE was resolved by about 5.5 grids in the spanwise direction. The measures of the domain were identical with those of the sloping section of the laboratory wave flume (see FIG. 3 a). The analytical velocity, pressure, and surface elevation on the basis of the second-order cnoidal wave theory are given to generate regular progressive waves at the offshore boundary. The LES results depend on the turbulent boundary conditions. While a wall law based on a logarithmic velocity profile has been used for computations of turbulent boundary layer flows, a primitive non-slip condition has been applied to wave breaking flows [38, 39, 4, 40], since the breaking-wave-induced turbulence transported to the near-bed region disrupts the boundary layer flow and thus the logarithmic profile is no longer acceptable there. We also employed the non-slip impermeable boundary condition at the bottom. A periodic condition
PT
315
ED
M
310
0.
AN US
300
=
CR IP T
295
used to model the sub-grid stress τ . All variables are non-dimensionalized by the water depth, gravity, and water density. The level-set method (Osher and Fedkiw [36]) was introduced to compute the free-surface form. In this method, a level-set function φ, representing the signed distance from the surface, defined the surface location at φ(x, t) = 0, and followed the advection equation;
AC
325
330
19
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 10: Evolution of the breaking wave surface and iso-surface of the dye concentration (c∗ = 0.2), (left), and the structures of the vortex cores (right) at an early stage of wave-breaking in case 3; the vortex cores for primary rollers with spanwise vorticity are colored green, and those for vortex loops with a counter-rotating vorticity are colored red for positive streamwise vorticity and blue for negative streamwise vorticity. See also supplementary movie 5.
20
ACCEPTED MANUSCRIPT
355
360
CR IP T
c∗ = c∗0 ea
AC
CE
365
AN US
350
M
345
ED
340
was applied at the side boundaries. We confirmed that the wave field in the surf zone achieved a quasi-steady state after the third wave-breaking occurred from the initial still water state; that is, the breaking-wave height and depth identical to the experimental measurements (see Table 1) were attained for the third breaking waves. While the current single-phase (liquid phase) computation excludes the dynamics of the settling solid sediment, a fraction of the numerical dye was computed as the simplest model of the volume concentration of passive sediment, which may approximately describe dilute particle-laden flows containing very fine sediment without contact stress, characterized by a small Stokes number. It should be noted that Stokes number for the experimental conditions in §3, ρs d250 p g St = 18µ h ≈ 0.025 1, indicating passive particle behaviors to the carrier flows. The dye concentration, c∗ , was defined in 0 ≤ c∗ ≤ 1, and governed by the advection equation Dc∗ = 0. (12) Dt Since equation (12) excludes the effects of turbulent diffusion and settling, the dye is passively advected without approaching an equilibrium state that the settling sediment is kept in suspension by the turbulent diffusion. In general, the turbulent diffusivity of a solid particle, described in terms of Schmidt number (Sc ), varies in gravity environment due to inertia of the particle, depending on relative response time to turbulence time-scale [41, 42]. Sc has been empirically determined for trivial flows, such as Sc =0.7 [43] and 0.56 [44] for open channel flows, 0.8 for flow bellow a pipeline [45], and variable Sc (0.56 – 0.7) for oscillatory flows[15]. However no available Sc model has been proposed for the breaking-wave-induced turbulence with wide-ranging fluctuation timescales. Since a coherent formation of sediment clouds, observed in FIG. 2, may be caused by the advection transport by organized flows not simple diffusion process, we concentrated on computing the governing advection effect to the coherent suspension by equation (12) without any bias associated with uncertain Sc during the wave breaking process. In this computational test, the advection effect of vortices to the initial suspension behavior of near-bed sediment is considered as a deviation from the equilibrium concentration typically observed in the physical experiments. Analogous to equation (6), the initial exponential profile of the concentration was assumed to be in an equilibrium state;
PT
335
370
∗ ∗
z
,
(13)
where the reference concentration c∗0 =1, the attenuation rate a∗ ≈ −30, and z ∗ is the normal coordinate from the bottom boundary (see FIG. 9). This profile was initially given for the entire surf zone when the third wave commenced breaking. It should be noted that the organization of dye transport is independent on the initial c∗ profile since c∗ is passively transported by the local fluid flow, following equation (12). Therefore the dye quantity never affects the carrier flow. 21
ACCEPTED MANUSCRIPT
380
While the current single-phase LES model has been validated for breaking waves [4] and for liquid impacts to free-surface [9], the identical model without a sediment model was used in the computational test, assuming minor sediment effects at an early stage of the suspension process when the breaking-waveinduced turbulence arrives the near-bed region. The validity of the current model assumptions at the early suspension process is discussed in Appendix.
CR IP T
375
3.2. Vortex structures
390
CE
405
PT
400
ED
M
395
AN US
385
While vortex dynamics analogous to Watanabe et al. [4] were observed in the current computation, distinct features of the near-bed transport process were identified in the computations. FIG. 10 shows the evolution of the surface forms, the iso-surfaces of the dye concentration, and the distributions of the vortex cores defined by the λ2 method (Jeong and Hussain [46]), (see also supplementary movie 5). After wave plunging (FIG. 10 a), a primary horizontal roller vortex (colored green) is initially produced around the inner surface of an overturning jet. The vortex loops with streamwise counter-rotating vorticity (colored red and blue for positive and negative vorticity, respectively) are then formed between the preceding secondary jet and the roller vortex (FIG. 10 b). The upstream and downstream bends of the vortex loops are wrapped by each primary roller and thus stretched in the oblique direction to organize a rib structure of counter-rotating vortices between them (FIG. 10 c). Because the counter-rotating vortices entrain fluid downward from the free-surface into the inner water mass to form longitudinal scars along the vortex axes[6, 7], transverse surface deformation is observed behind the wave front (see supplementary movie 5, and also Watanabe and Mori [8]). We find that transverse fluctuations of the dye concentration also occur when the upstream bends of the counter-rotating vortices approach the bed. The fluctuations of the dye concentration are amplified according to the evolution of the three-dimensional vortex structures (see FIG. 11 and supplementary movie 5). Zhou et al.[11] computationally observed that velocity fluctuations caused by the ODEs are associated with sediment suspension on a barred beach. They found no suspension occurred between the cores of ODEs where the downward vertical velocity fluctuation exited, while the sediments were advected upward on the side of the ODE with upward velocity fluctuation. The current results that the dye suspensions locally occur at the transverse locations where the vortex loops are wrapped by the upstream primary roller (see FIG. 11 b and c) is consistent with the finding by Zhou et al.[11], which supports the current dye model describes major features of the initial suspension event govern by ODEs. Now we discuss how the orientation of the vertical flows alternates around ODEs and what is the mechanical role of ODEs to the near-bed sediment transport, in the following sections.
AC
410
415
3.3. Dye transport in vortex field FIG. 12 shows the contours of the streamwise vorticity and the dye concentrations on a (y ∗ - z ∗ ) cross-section at x∗ = 12.77 where the maximum rise
22
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 11: Evolution of the breaking wave surface and iso-surface of the dye concentration (c∗ = 0.2), (left), and the structures of the vortex cores (right) during the splash-up process in case 3; the vortex cores for primary rollers with spanwise vorticity are colored green, and these for vortex loops with counter-rotating vorticity are colored red for positive streamwise vorticity and blue for negative streamwise vorticity. The phases (a) - (c) and the rectangle cross-section at x∗ =12.77 correspond to FIG. 12. See also supplementary movie 5.
23
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 12: Contours of the streamwise vorticity (ωx ) and the dye concentration (c∗ ) on a transverse-vertical (y ∗ − z ∗ ) plane at x∗ = 12.77 (see the rectangle cross-section in FIG. 11). The phases (a) - (c) correspond to FIG. 11. See also supplementary movie 6
.
24
ACCEPTED MANUSCRIPT
430
435
CE
450
PT
445
ED
M
440
CR IP T
425
AN US
420
of dye was observed, at the corresponding phases in FIG. 11 (see also supplementary movie 6). The transverse wavy distributions of the dye concentration appear near the bed when the descending vortices approach the bed (see FIG. 12 a). The fluctuations are amplified beneath the counter-rotating vortices where indicated by black arrows in FIG. 12 b, and then the concentrated dye is vertically ejected between the vortices and extended up to the surface level (see FIG. 12 c and supplementary movie 6). This suspension process of the dye may be interpreted in terms of vorticity dynamics. The convergent near-bed flow is induced at the convergent points, indicated by black arrows in FIG. 12 b, beneath the vortex pair with clockwise (positive) vorticity located on the right side of the convergent point and counterclockwise (negative) vorticity on the left side (see also FIG. 15). This flow induces transverse dye transport toward the convergent point from both sides, resulting in regular patterns of depositions in the transverse direction (FIG. 12 a). The deposited dye is carried by upward flows induced between these vortices and ejected up to the free surfaces (see FIG. 12 b and also FIG. 15). In contrast, the counter-rotating vortices with the opposite vorticity orientation induce divergent near-bed flows to sweep the dye away from the divergent points (red arrows in FIG. 12). As the ejected dye is consecutively carried upward by the vortex-induced flows, a rapid increase in the dye concentration is observed over the depth (see supplementary movie 6). Although the computation shows the passive dye transport without settling effects, the identical contributions of the counter-rotating vortices to the initial suspension process may cause an analogous upward sediment transport, which is again discussed in §3.4. Once the sediment is suspended, the concentration may be modified by the settling effect, which may differ from the computed dye motion at a later stage. FIG. 13 compares the typical sequential images of experimental breaking waves, and the computed vortex cores and dye concentration at y ∗ = 1.2 where the maximum dye rise was observed (see FIG. 12). Assuming air bubbles are entrapped within the upper parts of ODEs during wave plunging (FIG. 13 a– b) and are transported to depths with the descending vortices (FIG. 13 c – e), the computed vortex cores present consistent displacements with the observed ODEs during passage of the breaking wave (FIG. 13 f – j). When the computed vortex cores arrive the seabed, the bottom layer with high dye concentration becomes thicker near the offshore ends of the cores (at x ≈ 60 – 65 cm). The upward ejection and overturning transport of the dye can be then observed near x ≈ 65 cm (corresponding to x∗ ≈ 12.77 in FIG. 11) where the spanwise roller vortex wraps rear ends of the streamwise vortices (FIG. 11) and the vertical dye ejection occurs between the vortices (FIG. 12). Munro et al. [47] experimentally studied sediment resuspension and erosion induced by a vortex ring interacting with a sediment layer. They found the near-bed sediment is displaced in a high shear region bellow the ring core, resulting in the formation of a circular crater mound with spokelike scars (azimuthal deformation) in the crater interior. The azimuthal array of the radial deposition and erosion was concluded to be caused by the local sediment displacement owing to the azimuthal velocity induced by the azimuthal instability
AC
455
460
25
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 13: Sequential snapshots of the experimental suspended sediment at the plunging point, (a) – (e), and the computed vortex cores and dye concentration at y = 1.23, where the maximum dye rise is achieved, at the corresponding phases (f ) – (j); the phase intervals of (a) – (d) and (e) – (h) are 0.156 s.
26
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
Figure 14: Schematic illustration of the sediment suspension by the vortex ring; the divergent and convergent near-bed flows induced by the counter-rotating azimuthal vortices above the bed may cause bed load sediment transport toward the converging locations where the sediment is carried by upward flows induced between a pair of the counter-rotating vortices. The suspended sediment may be then carried by the primary vortex flow and rolled up around the ring perimeter.
27
ACCEPTED MANUSCRIPT
475
480
M
485
CR IP T
470
of the vortex ring[48]. Watanabe et al. [33] computationally found that the azimuthal instability results in a wavy deformation of the primary vortex ring and the formation of counter-rotating radial vortices wrapped by the vortex ring. FIG. 14 illustrates the possible mechanism of azimuthal suspension and deposition by the unstable vortex ring wrapping counter-rotating vortices approaching the sediment bed. The counter-rotating radial vortices wrapped by the primary vortex ring may induce divergent and convergent azimuthal nearbed flows beneath each vortex pair along the ring axis, resulting in the azimuthal array of the spokelike sediment depositions and erosions in the crater. The suspended sediments are then carried by the primary vortex flow and are rolled up around the ring perimeter, resulting in coherent rolled-up sediment clouds toward the crater interior[47]. The suspension and deposition processes by the unstable vortex ring provide analogies with those by the breaking-wave-induced vortices. In the wave breaking case, the dye lifted between the counter-rotating vortices from the depositions (FIG. 12) are rolled up by the primary roller to form the overturning dye pattern around the roller vortex (FIG. 13 right). The similar patterns of rolled-up sediments are also observed in the experiments (see FIG. 13 (c) – (e), and also FIG. 2 (c) and (d)). The high sediment concentration in 60 ≤ x ≤ 100 cm, observed in FIG. 6, may be interpreted by the effects of the suspension around the primary horizontal roller wrapping the streamwise vortices via the combined mechanisms of the vortex induced flows by the counter-rotating vortices and the primary roller.
AN US
465
ED
CE
495
PT
490
3.4. Summary of computed results FIG. 15 schematically illustrates the mechanical contributions of the counterrotating vortices to the suspension process of the sediment. The counter-rotating vortices, produced between the primary rollers, are stretched obliquely downward, and are achieved at the sea bed. The convergent and divergent flows are induced over the bed underneath the counter-rotating vortices, resulting in the transverse bed-loads of sediment and thus the transverse patterns of local depositions at the convergent points on the bed. The deposited sediment is then carried by the upward flow induced between the counter-rotating vortices, causing local suspension, as observed in FIG. 2 and FIG. 13. Since the suspended sediment is rapidly ejected and rolled-up around the primary roller, the high sediment concentration is maintained there, as observed in FIG. 5 and FIG. 6. It should be noted that statistically defined turbulent energy may contain effects of large-scale fluctuations of main flows of waves and organized vortices as well as small-scales turbulence directly produced by wave-breaking and seabed boundary layer flows, since the statistical averaging cannot separate turbulence scales. That is, the experimental turbulent energy in FIG. 4, FIG. 5 and FIG. 6 cannot be clearly separated into each contribution of the factors through the wave breaking process. However the large-scale coherence of the observed sediment clouds (FIG. 2 and FIG. 13) are unlikely to be caused by the simple diffusion process due to small-scale turbulence. The analogies of the suspension processes by vortex rings [47] provide the consistent interpretation on the contribution of the vortex-induced flows to suspension process (§3.4). Therefore we conclude that
AC
500
505
28
ACCEPTED MANUSCRIPT
510
the coherent vortex-induced flow is one of the major factors to suspend the sediment. This mechanism to induce sediment suspension in the three-dimensional vortex organization is introduced in a suspension model in the next section.
520
525
In §2, the previous suspension models for wave-breaking turbulence were observed to predict inconsistent sediment concentrations during the experimental breaking process. In §3, the vortex-induced suspension has been identified as a major mechanism excluded in the previous models. In this section, we propose a new approach to parameterize the vortex-induced sediment suspension. This approach analytically describe the vertical velocity induced by streamwise counter-rotating vortices in terms of vortex dynamics. The derived model is introduced into the diffusion-advection equation through parametric relations of the transverse flows and wave breaking conditions (Otsuka and Watanabe [12]).
AN US
515
CR IP T
4. vortex-induced suspension model
4.1. Modification to the advection-diffusion equation of sediment concentration A general form of mass conservation for volume-averaged suspended particles in liquid-solid two-phase flow is written by (14)
M
∂ ρs c + ∇ · (ρs cus ) = 0, ∂t
n
n
n
where us,n and Vn are the velocity and volume of the n-th particle, respectively. Equation (14) for the constant ρs is transformed into
PT
530
ED
where ρs is the density of sediment particle. The concentration is defined as the volume fraction of N particles contained in the reference volume V0 , c = X N N N X X V0−1 Vn , and the mean particle velocity us = (us , vs , ws ) ≡ us,n Vn Vn , ∂ c + ∇ · (c us ) = −∇ · c0 u0s ∂t
(15)
CE
via Reynolds decomposition based on the ensemble average. By introducing a gradient diffusion model, c0 u0s ≈ −s ∇c, to equation (15), we obtain (16)
Assuming that the horizontal gradients in c are negligibly smaller than the vertical gradient and time-averaging equation (16), the convection-diffusion equation for temporal mean concentration hci is approximated by
AC 535
∂ c + ∇ · (c us ) = s ∇2 c ∂t
∂ ∂ 2 hci hcihws i = hs i , ∂η ∂η 2
(17)
where η is the vertical coordinate with the origin at the undisturbed bed (η = z + h, see definitions in FIG. 3 b). If only the terminal settling velocity is 29
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 15: Schematic illustration of the mechanism of sediment suspension by the vortexinduced flows under breaking waves; the divergent and convergent near-bed flows induced by the counter-rotating streamwise vortices above the bed cause bed load sediment transport toward the converging locations where the sediment is carried by upward flows induced between a pair of the counter-rotating vortices. The suspended sediment may be then carried by the primary vortex flow and rolled up around the roller perimeter.
30
ACCEPTED MANUSCRIPT
545
∂ ∂ 2 hci . hci (hwi − wf ) = hs i ∂η ∂η 2
(18)
Equation (18) describes the explicit contribution of the vertical fluid flow to pick up sediment to balance the settling convection and the turbulent diffusion. Assuming the constant turbulent mixing by breaking wave hs i, as explained in §1.3, the exponential solution for equation (18) is thus derived as
AN US
550
CR IP T
540
considered as the sediment particle velocity, hws i ≈ wf , assuming a steady mechanical balance between gravitational particle settling in still water and spherical particle drag, ρCd wf2 = 4aρs g/3, equation (17) becomes equivalent to the conventional convection-diffusion equation (1). In the steady state of particle motion with velocity hws i in fluid flow with velocity hwi, the particle drag (in proportion to the squared relative velocity) balances with the gravity; 2 ρCd (hwi − hws i) = 4aρs g/3, where Cd is the drag coefficient. Therefore, as2 suming steady particle motion, we get (hwi − hws i) = wf2 , leading the relation hws i = hwi − wf . Equation (17) is now rewritten as
hci = c0 e−
wf −hwi η hs i
.
(19)
M
It should be noted that this solution excludes effects of the cross-shore sediment transport by the undertow and the rolled-up flow induced by the horizontal roller (FIG. 13), which may cause uncertainty of the prediction, as discussed again in §4.5. 4.2. Vortex induced transporting velocity
560
Let us consider N point vortices located at (yi , zi ) with circulation Γi on a (y − z) two-dimensional plane (vortex number i=1, 2, . . ., N , see FIG. 16 a) as a model of a streamwise vorticity field on a transverse-vertical cross-section (as shown in FIG. 12). The velocity (V , W ) induced by these vortices at arbitrary locations (y, z) is determined on the basis of the well-known Biot-Savart integral [49]: N X Γi z − zi V (y, z) = − (20) 2 2 2π (y − yi ) + (z − zi ) i=1
AC
CE
PT
ED
555
565
W (y, z) =
N X Γi y − yi 2π (y − yi )2 + (z − zi )2 i=1
(21)
Assuming a pair of counter-rotating vortices with identical absolute circulation, i.e., Γ = Γ1 = −Γ2 and N =2, are horizontally aligned at the identical vertical level z0 = z1 = z2 (see FIG. 16 b), equations (20) and (21) are reduced to ! z − z0 z − z0 Γ − (22) V =− 2 2 2π (y − y1 )2 + (z − z0 )2 (y − y2 ) + (z − z0 )
31
AC
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 16: Schematic representations of (a) streamwise point vortices located in a transversevertical cores-section, as a model of a vorticity field in FIG. 12, (b) a pair of counter-rotating vortices with identical absolute circulation Γ, located at z = z0 ., and (c) upward induced flow flux for a pair of counter-rotating vortices with spacing λ. Wc indicates the induced velocity at the center of the vortex pair.
32
ACCEPTED MANUSCRIPT
Γ W = 2π
y − y1 2
(y − y1 ) + (z − z0 )
2
−
y − y2 2
2
(y − y2 ) + (z − z0 )
!
(23)
Wc =
575
where λ is the distance of the vortices. If these are assumed to be Rankine vortices with vorticity ω, the circulation at the edge of the vortex core is given by Γ = 2πrc2 ω = 2πrc γ [6], which is used as the representative circulation of the point vortices, where rc is the core radius and the tangential velocity γ = rc ω.
4.3. Model calibration Otsuka and Watanabe [12] experimentally measured the transverse flows induced by the counter-rotating vortices by using an ultrasonic velocity profiler, and statistically parameterized the characteristic length (λ), time (τ ), and velocity (γ) of the transverse flows in breaking waves. These characteristic values exhibit in-phase variations; the shortest length and time scales are achieved at the wave front, and they gradually increase to become the longest length and time scales behind the wave crest. If these scale properties are assumed to be maintained near the bed, the distance between the vortices, the duration that the flow is influenced by the vortices, and the tangential velocity of the vortices can be approximated by the dimensionless empirical parameters [12]; the maximum length λ/S ≈ −0.64Fr + 0.61, the maximum √ time τ /T ≈ −0.77Fr + 0.66, and the maximum transverse velocity Fr = γ/ gh ≈ 0.47ξ + 0.09, where T is the wave period, ξ is the surf-similarity parameter (5) and the reference length S = 0.25 (m). Here λ/S, τ /T and Fr are physically interpreted as the relative transverse spacing of the counter-rotating flows to the flume width, the relative duration of the transverse flow to the wave period, and the absolute transverse velocity with respective to the wave celerity, respectively. Otsuka and Watanabe[12] compared the spacing between the counter-rotating flows observed in two wave flumes with different width (S), and found λ is independent on S for the same wave-breaking conditions; that is, the organization of the counter-rotating vortices is inherently determined by the breaker conditions regardless of the flume width. It should be noted that the above empirical parameters are available within the range of 0.20 ≤ ξ ≤ 0.73 in the cases of regular wave breaking over an uniform beach slope. Substituting λ and γ into equation (24), Wc at the bottom, z = −h, is parameterized by
CE
590
PT
585
ED
M
580
(24)
AN US
570
λ 2Γ , 2 π λ + 4 (z − z0 )2
CR IP T
The transverse velocity, V , is canceled out at the center of the point vortices, y = (y1 + y2 ) /2 , while the induced vertical velocity there, Wc , is given by
AC
595
600
Wc ≈ 4rc
p
ghFr
S (−0.64Fr + 0.61) 2
2,
S 2 (−0.64Fr + 0.61) + 4 (−h − z0 )
(25)
where the parameterized Froude number Fr = 0.47ξ + 0.09. The core radius, rc , may depend on a breaker. In this study, it was estimated to be rc ≈ 0.01 33
ACCEPTED MANUSCRIPT
620
625
CR IP T
615
AN US
610
M
605
m based on the computed results. The unknown travel depth of the vortex is assumed as z0 ≈ −H to be consistent with the computed one. Equation (25) now indicates the parameterized characteristic velocity to vertically carry the sediment beneath the organized vortex structures of breaking waves. The parameters rc and z0 may depend on breaker conditions, especially breaking wave height and depth. These empirical parameters may need to be redefined for larger scale experiments or in-situ observations. Now we consider the mean upward carrier flow contributing upward sediment flux from the near-bed region. As already discussed, when the counter-rotating vortices approach the sea bed, the bottom sediment is driven by the near-bed convergent flow for being deposited and ejected upward between the vortex pair, while inner water mass is transported downward between the next vortex pair with opposite vorticity orientation to create divergent near-bed flow sweeping the sediment away (see FIG. 12 and FIG. 15). In an equilibrium state of sediment concentration following the exponential distribution, the near-bed concentration may be much higher than the upper one away from the bed, and thus the upward induced flow gives a major contribution to the net vertical sediment flux. In the simplest case of clear water (with no suspended sediment) above the vortices, the downward induced flow provides no contribution to the vertical sediment transport and thus the net sediment flux is defined only by the upward vertical transport. Assuming that the upward carrier flow flux between the vortex pair Ry is given by y12 W dy ≈ Wc λ (see FIG. 16 c), the temporal mean of the upward induced carrier velocity, hWc i, per unit width containing a pair of the counterrotating vortices (2λ) may be approximately given by Wc Wc τ ≈ (−0.77Fr + 0.66) , 2 T 2
(26)
ED
hWc i ≈ α
PT
CE
630
where α is the empirical parameter representing a ratio of sediment concentrations in a near-bed region and far from the bed; α = 1.0 for the clear water case, and α = 0.0 for the case of complete sediment exchange (no net sediment flux) in the uniform vertical profile of concentration. In this study, α =1.0 is used for simplicity. Identifying hwi with the time-averaged upward fluid flow induced by the counter-rotating vortices at the bed, hWc i, equation (19) is rewritten by hci = αc0 e−
640
.
(27)
Equation (27) provides the mechanical contributions of the vortex-induced flows, computationally supported in §3, to the vertical distribution of suspended sediment, experimentally observed in §2. Following Larson and Kraus [21], equation (8) is used as the mixing coefficient hs i for the turbulence caused by wave breaking in this study. FIG. 17 shows the cross-shore distributions of the predicted vortex-induced velocity for the experimental wave cases. It should be noted that the model assumption is acceptable in the shoreward region from the wave-plunging location (x ≥ 0.6 m) where the counter-rotating vortices begin to be produced after
AC 635
wf −hWc i η hs i
34
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 17: Cross-shore variations of parameterized mean vertical velocity induced by the counter-rotating vortices, hWc i, for the experimental cases.
35
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ED
Figure 18: Comparisons of the attenuations rate predicted by the current model and Larson and Kraus [21]; case 1 (red), case 2 (blue), and case 3 (black).
PT
CE
645
wave plunging. The sediment receives vertical drag in the induced flows with hWc i against the mean settling velocity wf =0.0248 m/s. hWc i slightly slower than wf are maintained in the transition region (0.6 m≤ x <1.2 m), and gradually decrease during the wave-breaking process; that is, the upward drag due to hWc i significantly decelerates the mean settling behaviors of the suspended sediment in the transition region, while this effect is reduced in the bore region. In the spilling wave-breaking case (case 1), the induced velocity is smaller than the plunging-wave cases (cases 2 and 3) over the entire surf zone because the effects of the vortices are smaller in this case, resulting in less suspension (see FIG. 4).
AC
650
4.4. Model validation with experimental data
655
FIG. 18 compares the current attenuation model hac i = − (wf − hWc i) /hs,lk i with alk by Larson and Kraus [21] in addition to the experimental results. Because our model uses a mixing coefficient identical to that used by Larson and Kraus [21], equation (8), the difference between the models explicitly indicates 36
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
ED
Figure 19: Comparisons of the predicted attenuation rate predicted by the current and previous models with the experimental results; case 1 (red), case 2 (blue), and case 3 (black).
PT
CE
660
the quantitative contribution of the vortex-induced flows to the sediment suspension. The current model ac for the plunging-wave cases (cases 2 and 3) in the transition region are consistent with the experimental attenuation rates, and thus the prediction is significantly improved over that provided by the method of Larson and Kraus [21] predicting much larger attenuation, which suggests that vortex-induced flow is an important factor for describing the concentration of the suspended sediment in the surf zone. The improvement of the prediction for spilling waves producing weaker vortices only near the free surfaces, which may violate the current model assumptions, however, is inadequate. It should be noted that, as the current model assumes that the turbulent mixing is independent on z (constant turbulent mixing), the attenuation rate is also independent on the reference concentration c0 in our approach. FIG. 19 shows the predicted attenuation rates of the current model compared with the previous models by van Rijn [18], Larson and Kraus [21], and Goda [29]. As explained above, while the current model for plunging-wave cases describes attenuations consistent with experimental values, the predicted atten-
AC
665
670
37
ACCEPTED MANUSCRIPT
690
695
CR IP T
5. Conclusions
PT
700
Limitations of the proposed model and uncertainties of the prediction are discussed in this section. Since we followed the conventional assumption of negligibly small cross-shore variations in hci and solved one-dimensional advectiondiffusion equation for the mean concentration, any effects of cross-shore transport by the undertow and the rolled-up flow around the primary roller were excluded from the prediction. As the travel depth of the vortex, z0 , and the core radius, rc , used for determining the vortex-induced flow (equation (25)), may depend on a scale of breaker, these parameters should be reexamined through a parametric study for arbitrary wave-breaking cases. The parametric length, time and velocity sales of the transverse flows, proposed by Otsuka and Watanabe [12], are limited to use in the range of the experimental surf similarity parameter, 0.20 ≤ ξ ≤ 0.73, for regular wave-breaking over a uniform sloping breach. Statistical uncertainties of the near-bed concentration, observed in FIG. 7, may be relevant to uncertain strength of the breaking-wave induced turbulence and vortices as well as uncertain local location where they arrive on the bed. The other possible parameters which may affect sediment transport in the surf zone, such as mobility of bottom sediment, local deformation of seabed, irregularity of incident waves, are excluded in the model, which also needs to be taken into account for further improvement of the prediction in future study.
AN US
685
4.5. Limitation of model parameters
M
680
ED
675
uations are all much larger in the previous models. Wang [22] also compared the model predictions of vertical profiles of the sediment concentration in the surf zone and found the both models by van Rijn [18] and Larson and Kraus [21] underpredict the concentration, which may be due to the models overestimating the attenuation. The proposed model including the vortex-induced suspension mechanism, supported by experimental and computational evidence in §2 and 3, reasonably predicts the mean suspension concentration under violent breakers.
CE
In this study, physical and computational experiments were performed to determine the mechanical contributions of counter-rotating vortices that are sequentially produced during repeated splash-up cycles to sediment suspension in the surf zone. Experimental measurements of the surface elevation, sediment concentration, and fluid velocity distribution revealed that hight sediment concentration appears at neither the wave front following the shoreward velocity nor the region containing strong turbulence behind the wave crest. Vertical profiles of the observed time-averaged sediment concentrations are well defined by exponential attenuations regardless of the breaker type. Previous models fail to predict the measured attenuation rates, causing underpredictions of the sediment concentration in the surf zone, as also observed by Wang [22] and van Rijn [18]. These experimental results indicate another mechanism inducing suspension in addition to the turbulence caused by seabed boundary layers and breaking waves.
AC
705
710
38
ACCEPTED MANUSCRIPT
730
735
745
PT
ED
740
CR IP T
725
AN US
720
We investigated transitional concentrations of passively transported numerical dye, as the simplest model of dilute particle-laden flows involving very fine sediment, in breaking-wave-induced vortex structures in LES computational experiments. When vortex loops with streamwise counter-rotating vortices are stretched obliquely downward between the adjacent primary rollers and approach the bed, transverse dye transport driven by divergent and convergent near-bed flows are induced by the counter-rotating vortices over the bed, resulting in the formation of regular patterns of transverse depositions at the convergent points. The upward flows induced between the counter-rotating vortices carry the deposited dye and eject it upward. A rolled-up coherent pattern of the dye concentration is formed by the advection around the primary roller perimeter. Analogies with a vortex structure involving a vortex ring, wrapping an azimuthal array of the counter-rotating vortices, and the local suspension induced by the vortices [47] support our interpretation of the vortex-induced suspension under breaking waves. While this experiment omits gravitational settling behaviors of sediment, identical contributions of the counter-rotating vortices to the initial displacement and suspension may result in analogous upward ejections of sediment due to obliquely descending eddies, as observed in FIG. 2 and FIG. 13. In addition to the effects of settling and breaking-wave-induced turbulence, the mechanism to initiate suspension due to vortex-induced flows was incorporated into a suspension model in terms of vorticity dynamics with parametric characteristics of transverse flows in breaking waves. The proposed model reasonably predicts the observed exponential attenuation rates for violent breakers and significantly improves the prediction accuracy. Note that the validity of the proposed model is supported within the conditions of the current experiments and the previous parametric study[12]. The model includes uncertainties associated with cross-shore sediment transport, local erosion of seabed, breaker-dependent vortex cores and travel depth of the vortices.
M
715
Acknowledgments
CE
This research was supported by JSPS Grant-in-Aid for Scientific Research (15H04043).
AC
A. Comparison of computational and experimental turbulences
750
The turbulence statistics in the laboratory experiment is compared with the computed flows in this section. In the physical experiment, Reynolds decomposition, based on the ensemble averaging, was used to define the turbulence energy; k=
1 0 0 u ·u , 2 39
(28)
ACCEPTED MANUSCRIPT
770
775
CR IP T
PT
780
AN US
765
M
760
ED
755
where the velocity fluctuation u0 = u − u, and u is the ensemble-averaged velocity. On the other hand, in LES, the sub-grid turbulence energy, ksgs , is defined by 1 1 00g 00 e·u e) ≡ (29) u ·u , ksgs = (ug ·u−u 2 2 ˜ and u ˜ is the grid scale velocity. where the sub-grid scale velocity u00 = u − u While ksgs is the kinetic energy of small-scale turbulence with the length scale less than the grid width ∆ ≈ 6 mm, the statistical definition k may contain large-scale velocity fluctuations of main flows of waves and large scale vortices since the statistical average cannot separate turbulence scales. Therefore ksgs cannot be directly compared with k. The turbulence statistics based on the ensemble averaging for the LES results cannot be obtained straightforwardly since no variance has been computed in any trials with the the identical initial conditions. Therefore it was difficult to obtain the ensemble-averaged representation of the current LES results for fairly comparing with the experimental one. On the other hand, a phase-averaging operation over waves has been also used for estimating the flow statistics in breaking waves (e.g. Ting and Kirby[50], Cox and Kobayashi [51], Kimmoun and Branger [52]). Although the phase averaging may result in a certain difference in turbulence components from the ensemble one because of the a-periodic nature in breaking point and wave period [53], fundamental features of statistical flow variations estimated from the both definitions may be comparable in the regular breaking process. In order to estimate the statistical representation of our LES results, we performed the phase-averaging operation to the computed flows in the same manner as Watanabe et al.[4]. The instantaneous grid-scale velocity, b e , is redefined in terms of the phase-averaged grid-scale velocity u e by u e = u 000 000 b e +u e , where the superscript indicates the deviation from the phase-averaged u ˜ + u00 , we find u b = quantity. Taking the phase-average of the relation u = u 000 b c00 . As u = u e+u b + u , the turbulent energy based on the phase-averaged kp u is described as
CE
kp =
1 000d 000 1 000d 000 d 1 c00 c00 b c00 e ·u e + ksgs − u · u − u e·u . u ·u = u 2 2 2
(30)
AC
c00 is much smaller than the other terms, the above equation may be Assuming u approximately reduced to
785
kp ≈
1 000d 000 d e ·u e + ksgs . u 2
(31)
√ Following Zhou et al. p [11], the experimental turbulent intensity 2k and the computational one 2kp are defined to compare in FIG. 20, together with b and u. We find overall features of the temporal variations in the the velocity u computed velocity are consistent with the experimental ones, and however the computed velocity tends to overestimate with the wave propagation. The computed turbulent intensity agrees with the experimental one at the early breaking 40
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
Figure 20: Time records of computed phase-averaged results (solid line) and experimental ensemble-averaged ones (broken line) at η=4.5 cm; dimensionless horizontal velocity (top), vertical velocity (middle) and turbulent intensity (bottom).
41
ACCEPTED MANUSCRIPT
800
805
CR IP T
795
process (x = 0.4 and 0.6 m). In the later stage, while the overestimations are observed at x = 0.8 and 1.1 m, the both intensities present the rapid increase from the identical phases; that is, the inception of the computed turbulent generation coincides with the experimental one. Since the current model assumes a single-liquid-phase flow, any effects of sediment have been excluded from the computation. On the other hand, the experimental incident wave energy should be spent for lifting sediment up [23]. In addition, the experimental flow field may be modified by multiple effects of suspended sediment, including additional dissipation due to particle drag, generation of additional turbulence, particle settling, and mobility of seabed. These effects may be associated with the overestimations of the computed velocity and turbulent intensity in x ≥ 0.8 m where may occur successive modifications of flows during the breaking process. In order to precisely compute the turbulent flow with sediment throughout the surf zone, appropriate modeling of sediment-turbulence interaction is required. However the computed flow statistics at the early stage of wave breaking presents major features of the experimental turbulence, indicating the validity of the current computation under the assumption of minor effect of sediment at the early breaking process.
AN US
790
References
[1] D.H. Peregrine, Breaking waves on breaches, Annu. Rev. Fluid Mech. 15 (1983) 149–178.
M
810
815
ED
[2] I.A. Svendsen, P.A. Madsen, J.B. Hasen, Wave characteristics in the surf zone, Proc. 16th Coastal Engineering Conf. (1978) 520–539. [3] K. Nadaoka, M. Hino, Y. Koyano, Structure of the turbulent flow field under breaking waves in the surf zone, J. Fluid Mech. 204 (1989) 359–387. doi:10.1017/S0022112089001783.
PT
[4] Y. Watanabe, H. Saeki, G. J. Hosking, Three-dimensional vortex structures under breaking waves, J. Fluid Mech. 545 (2005) 291–328. doi:10.1017/ S0022112005006774. [5] T. Sarpkaya, Vorticity, free surface, and surfactants, Annu. Rev. Fluid Mech. 28 (1996) 83–128. doi:10.1146/annurev.fl.28.010196.000503.
CE 820
AC
[6] T. Sarpkaya, P. Suthon, Interaction of a vortex couple with a free surface, Experiments in Fluids. 11 (1991) 205–217.
825
[7] M. Brocchini, D. H. Peregrine, The dynamics of strong turbulence at free surfaces. part 1. description, J. Fluid Mech. 449 (2001) 225–254. doi: https://doi.org/10.1017/S0022112001006012. [8] Y. Watanabe, N. Mori, Infrared measurements of surface renewal and subsurface vortices in nearshore breaking waves, J. Geophys. Res. Oceans 113 (2008) C07015. doi:10.1029/2006JC003950. 42
ACCEPTED MANUSCRIPT
835
840
[9] A. Saruwatari, Y. Watanabe, D. M. Ingram, Scarifying and fingering surfaces of plunging jets, Coastal Eng. 56 (2009) 1109–1122. doi:http: //dx.doi.org/10.1016/j.coastaleng.2009.08.007. [10] P. D. LeClaire, F. C. Ting, Measurements of suspended sediment transport and turbulent coherent structures induced by breaking waves using two-phase volumetric three-component velocimetry, Coastal Engineering 121 (2017) 56–76. doi:http://dx.doi.org/10.1016/j.coastaleng. 2016.11.008.
CR IP T
830
[11] Z. Zhou, T.-J. Hsu, D. Cox, X. Liu, Large-eddy simulation of wave-breaking induced turbulent coherent structures and suspended sediment transport on a barred beach, J. Geophys. Res. Oceans, 122 (2017) 2017–235 doi: 10.1002/2016JC011884.
AN US
[12] J. Otsuka, Y. Watanabe, Characteristic time, length and velocity scales of transverse flows in the surf zone, Coastal Eng. J. 57 (02) (2015) 1550006. doi:10.1142/S0578563415500060. [13] H. Rouse, Fluid mechanics for hydraulic engineers, Dover, New York, 1961.
[15] L. Amoudry, T.-J. Hsu, P. L.-F. Liu, Schmidt number and near-bed boundary condition effects on a two-phase dilute sediment transport model, Journal of Geophysical Research, 110, C09003 (2005). doi:10.1029/ 2004JC002798.
ED
850
[14] L. C. van Rijn, Sediment transport, Part II: Suspended load transport, Journal of Hydraulic Engineering, 110 (11) (1982) 1613 – 1641.
M
845
[16] H. Lundgren, Turbulent currents in the presence of waves, Coastal Engineering Proceedings 12 (1970) 623–634.
[18] L. C. van Rijn, Principles of sediment transport in rivers estuaries and coastal seas, Aqua publications, 1993.
CE
855
PT
[17] I. G. Jonsson, Measurements in the rutbulent wave bounrary layer, IAHR 10th Congr. 1 (1963) 85–92.
AC
[19] P. Nielsen, Field measurements of time-averaged suspended sediment concentrations under waves, Coastal Engineering 8 (1) (1984) 51–72. doi: http://dx.doi.org/10.1016/0378-3839(84)90022-X.
860
865
[20] P. Nielsen, Three simple models of wave sediment transport, Coastal Engineering 12 (1) (1988) 43–62. doi:http://dx.doi.org/10.1016/ 0378-3839(88)90014-2.
[21] M. Larson, N. Kraus, Estimation of suspended sediment trapping ratio for channel infilling and bypassing, US Army Corps of Engineers, Technical Notes ERDC/CHL CHETN-IV-34 (2001) 1–9.
43
ACCEPTED MANUSCRIPT
[22] P. Wang, W. Yuan, L. Min, Measuring and modeling suspended sediment concentration profiles in the surf zone, Journal of Palaeogeography 1 (2) (2012) 172–192. doi:http://dx.doi.org/10.3724/SP.J.1261. 2012.00013.
875
[23] Y. Goda, Empirical formulation of sediment pickup rate in terms of wave energy flux dissipation rate, Coastal Engineering Journal 55 (04) (2013) 1350012. doi:10.1142/S0578563413500125.
CR IP T
870
[24] H. Gotoh, Accurate particle method for computational sediment dynamics, Proc. two-phase modelling for sediment dynamics in geophysical flows THESIS-2016 (2016) 3 –6.
880
AN US
[25] T.-J. Hus, Z. Cheng, J. Chauchat, X. Yu, A multi-dimensional multiphase sediment transport modeling framework - an open-source community modeling effort, Proc. two-phase modelling for sediment dynamics in geophysical flows THESIS-2016 (2016) 7 –10. [26] J. Fredsøe, R. Deigaard, Mechanics of Coastal Sediment Transport, Vol. 3 of Advanced Series on Ocean Engineering, World Scientific, 1992.
[27] J. J. Bosman, H. J. Steetzel, Time and bed averaged concentrations under waves (1986) 986–1000.doi:10.1061/9780872626003.074. [28] L. C. van Rijn, A. Kroon, Sediment transport by currents and waves, Coastal Engineering Proceedings (1993) 2613–2628.doi:10.1061/ 9780872629332.199.
M
885
[30] J.A. Battjes, Surf Similarity , Proc. 14th Int. Conf. on Coastal Engineering (1974) 466–480.
PT
890
ED
[29] Y. Goda, Re-evaluation of sediment pickup coefficient in the surf zone based on field measurement and large-scale experiment data , ECOH/YG technical report No.11 (2010), in Japanese.
CE
[31] C.E. Willert, M. Gharib, Digital particle image velocimetry , Experiments in Fluids 10 (1991) 181–193. [32] P. Lubin, S. Glockner, Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments, J. Fluid Mech. 767 (2015) 364–393. doi:https://doi.org/10.1017/jfm. 2015.62.
AC
895
900
[33] Y. Watanabe, A. Saruwatari, D. M. Ingram, Free-surface flows under impacting droplets, J. Comp. Phys. 227 (2008) 2344–2365. doi:http: //dx.doi.org/10.1016/j.jcp.2007.10.020. [34] M.S. Longuet-Higgins, Capillary rollers and bores, J. Fluid Mech. 240 (1992) 659-679.
44
ACCEPTED MANUSCRIPT
[35] V. Yakhot, S. A. Orszag, Renormalization group analysis of turbulence. I. basic theory, J. Sci. Comput. 1 (1) (1986) 3–51.
910
[36] S. Osher, R. P. Fedkiw, Level set methods: An overview and some recent results, J. Comp. Phys. 169 (2001) 463–502. doi:http://dx.doi.org/10. 1006/jcph.2000.6636.
CR IP T
905
[37] T. Yabe, T. Aoki, A universal solver for hyperbolic equations by cubicpolynomial interpolation I. one-dimensional solver, Comput. Phys. Commun. 66 (2-3) (1991) 219–232. doi:10.1007/BF01413793. [38] D. Lakehal, P. Liovic, Turbulence structure and interaction with steep breaking waves, J. Fluid Mech. 674 (2011) 522–577.
AN US
915
[39] P.D. Hieu, K. Tanimoto,, V. T. Ca, Numerical simulation of breaking waves using a two-phase flow model, Applied Mathematical Modeling 28 (2004) 983–1005. [40] T. Suzuki, A. Okayasu,, T. Shibayama, A numerical study of intermittent sediment concentration under breaking waves in the surf zone, Coastal Engineering 54 (2007) 433–444.
920
[41] S. Elghobashi, On predicting particle-laden turbulent flows, Applied Scientific Research 52 (1994) 309–329.
M
[42] C. T. Crowe, T. R. Troutt, J.N. Chung, Numerical models for two-phase turbulent flows, Annu. Rev. Fluid Mech. 28 (1996) 11–43.
[45] D. Liang, L. Cheng, F. Li, Numerical modeling of flow and scour below a pipeline in currents Part II. Scour simulation, Coastal Engineering 52 (2005) 43–62.
CE
930
[44] S. K. Jha, F.A. Bombardelli, Two-phase modeling of turbulence in dilute sediment-laden, open-channel flows, Environ. Fluid. Mech. 9 (2009) 237– 266.
PT
925
ED
[43] J. C. Winterwerp, Stratification effects by cohesive and noncohesive sediment, J. Geophys. Res. 106 (19) (2001) 22559–22574.
[46] J. Jeong, F. Hussain, On the identification of a vortex, Journal of Fluid Mechanics 285 (1995) 69–94. doi:10.1017/S0022112095000462.
AC
[47] R.J. Munro, N. Bethke, S.B. Dalziel, Sediment resuspension and erosion by vortex rings, Physics of Fluids 21 (2009), doi:10.1063/1.3083318 doi: 10.1063/1.3083318.
935
[48] E. Sheila, E. Windnall, J.P. Sullivan, On the stability of vortex rings, Proc. R. Soc. Lond. A. 332 (1973) 335–353. [49] S.E. Widnall, The structure and dynamics of vortex filaments, Annu. Rev. Fluid Mech 7 (1975) 141–165. 45
ACCEPTED MANUSCRIPT
940
[50] F.C. Ting, J.T. Kirby, Dynamics of surf-zone turbulence in a spilling breaker, Coastal Engineering 27 (1996) 131–160.
945
[52] O. Kimmoun, H. Branger, A particle images velocimetry investigation on laboratory surf-zone breaking waves over a sloping beach, J. Fluid Mech. 588 (2007) 353–397. [53] K.-A. Chang, P. L.-F. Liu, Experimental investigation of turbulence generated by breaking waves in water of intermediate depth, Phys. of Fluid 11 (1999) doi:10.106063/1.870198.
AC
CE
PT
ED
M
AN US
950
CR IP T
[51] D.T. Cox, N. Kobayashi, Identification of intense, intermittent coherent motions under shoaling and breaking waves, J. Geophys. Res. 105 C6 (2000) 14223–14236.
46