# An Experimental Study on Progressive and Reverse Fluxes of Sediments with Fine Fractions in Wave Motion

^{1}

^{2}

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## Abstract

**:**

_{i}< 0.20 mm, coarse, and total fractions all outgoing in the crest and trough phase from the initial area and deposited in adjacent control areas. As this model did not take into account both the effects of vertical mixing and the phase-lag effects related to the presence of fine and very fine fractions, a modification of this model was proposed that was based on four coefficients that corrected for fluxes. The consistency of the sediment transport calculations according to the modified model with measurements was achieved within plus/minus a factor of 2 of the measurements.

## 1. Introduction

_{net}defined for different grain mobility conditions. The operation of the wave-period-resultant sediment transport rate q

_{net}means that in the case of a symmetrical (sinusoidal) wave—when the transport rate of sediment outgoing from the initial area during the crest phase is equal in absolute value to the transport of sediment outgoing during the trough phase—the resultant transport q

_{net}will be equal to zero. This (q

_{net}= 0), in the absence of sediment flows feeding the area and coming from other areas, could suggest that the bottom level in the initial area remains unchanged. However, such a situation seems improbable and inconsistent with experiments [8], as it would imply that all grains mobilized in the crest and trough phases and carried away from the initial area return in the trough and crest phases to that area, respectively. For this reason, in the case of asymmetrical waves (acceleration-skewed oscillatory flows and those described by the Stokes approximation), it is difficult to expect that all grains of the flux outgoing in the crest phase and reduced by the magnitude of resultant transport q

_{net}during the wave period, as well as all grains carried away in the trough phase, will return to the initial area. This is because in all cases in which the magnitudes of sediment transport coming from other areas are zero, the changes in bottom level in the area under consideration are the result of sum of the absolute values of sediment transport outgoing from that area (in both directions) minus the magnitudes of sediment transport returning to that area. Therefore, the proportion of fluxes returning to the initial area relative to the outgoing fluxes, both in the wave crest and trough, remains an open question. The above issue becomes particularly important for sandy mixtures with fine and very fine fractions, as both velocity and sediment concentration not only change during the wave period, but are not necessarily in phase with each other. This means that the prediction of instantaneous and resultant sediment transport during the wave period is a very complex task.

_{50}= 0.21 mm showed a direction consistent with the wave direction, while that with a diameter of d

_{50}= 0.13 mm showed an opposite direction [14]. The reason for such sediment movement is the fact that small grains with a low falling velocity that were previously lifted from the bottom during the crest phase remain in suspension during the trough. As previously mentioned, the resulting phase shifts between maximum water velocity and maximum sediment concentration are particularly significant in cases in which the bottom is composed of sandy mixtures with a proportion of fine fractions (d

_{50}< 0.20 mm) [15].

- Models describing suspended sediment;
- Models describing bedload transport regime;
- Two- and three-layer models; i.e., describing both suspended and bedload transport; while a possible third layer describes the transition area between these layers and corresponds to the saltation layer.

_{i}< 0.20 mm), with a diameter di of the i-th sediment fraction.

_{i}< 0.20 mm. A multilayer description of sediment fluxes with a complete vertical structure of both instantaneous concentration and velocity was used for the widest possible range of grain mobility conditions. Therefore, it seems advisable to extend sediment transport studies for any grain size distribution, including sandy-gravel mixtures with fine and very fine fractions in the range of d

_{i}< 0.20 mm.

## 2. Materials and Methods

#### 2.1. Sediment Transport during the Wave Crest and Trough

_{j}of the control area, during the wave crest and trough, respectively, can be described by the equation:

^{+/−}(z,t) is the elementary instantaneous flux during the wave crest and trough, respectively, described as:

^{+/−}(z,t) and c

^{+/−}(z,t) are the instantaneous velocity and sediment concentration during the wave crest and trough, respectively.

- Integrated fluxes outgoing from the control area with the intensity of ${q}_{st}^{+/-}$ transporting sediment grains to adjacent control areas, in both directions. It is the part of integrated total flux with the intensity of ${q}^{+/-}$, which in the experiment described in this paper was retained in the traps on both sides of the control area. It was postulated [37] that the movement of a packet of grains z
_{k}in sediment fluxes ${q}_{st}^{+/-}$ follows horizontal projection trajectories of different but equally likely (with probability p = 1/K) lengths Δx_{k}, where:

- Integrated fluxes returning to the control area with the intensity of ${q}_{f3}^{+/-}$ transporting suspended sediment within the boundary layer of δ
_{m}thickness, defined according to the multilayer model [36,38]. These fluxes are the result of phase lags between water and sediment. These effects cause grains outgoing from the control area in the crest and trough phases to return and be retained in the control area by fluxes outgoing from the trough and crest phases;

- Integrated return fluxes ${q}_{f1}^{+/-}$ and ${q}_{f2}^{+/-}$ of sediments suspended high above the bottom and also above the boundary layer (${q}_{f1}^{+/-}$). Fluxes ${q}_{f1}^{+/-}$ and ${q}_{f2}^{+/-}$ result from the suspension of a large number of very fine sediment grain fractions above the bed, as a consequence of vertical sorting of the granulometrically heterogeneous sediment.

_{i}> 0.20 mm. Then, the following relation applies: ${({q}_{st}^{+/-})}_{j}={({q}^{+/-})}_{j}$. When the bottom is composed of sediments with a content of fine fractions (but without very fine fractions), phase lags cause that the flux-induced erosion of the bottom ${({q}_{st}^{+/-})}_{j}$ can be described (following [36]) by a flux $2\left|{q}^{-}\right|+{({q}_{net})}_{meas.}$, where the magnitude $\left|{q}^{-}\right|$ is proposed to be described by the Kaczmarek et al. 2022 model, while ${({q}_{net})}_{meas.}$ is the magnitude determined from measurements.

#### 2.2. Aim and Scope of the Study

_{i}< 0.20 mm, it neglected the effects of phase lags of fine fractions in this range, and did not describe the influence of these fractions on the transport volume ${q}_{st}^{+/-}$, so a modification of this model including the aforementioned effects was proposed.

#### 2.3. Experimental Investigation

#### 2.3.1. Experimental Setup

_{w}wave height.

#### 2.3.2. PIV System

#### 2.3.3. Sediment Transport and Grain Distribution Measurements

- Sandy with a very high proportion of fine fractions (d
_{50}< 0.20 mm; sand D: grain content d_{i}< 0.20 mm = 97%); - Sandy with low content of fine fractions (d
_{50}≥ 0.20 mm; grain content d_{i}< 0.20 mm: sand A = 6%, B = 10.5%, C = 26%).

## 3. Results and Discussion

#### 3.1. Theoretical Investigations

#### 3.1.1. Sediment Transport during the Wave Crest and Trough Phase

_{i}< 0.20 mm high above the bottom, as a result of vertical sorting of granulometrically heterogeneous sediment. Furthermore, it was assumed that the return flow ${q}_{f3}^{+/-}$ resulted from phase lags between water and fine fractions in the range of d

_{i}< 0.20 mm. A theoretical analysis should be based on a three-layer heterogeneous sediment model [36]. However, this model does not take into account the effects of additional vertical sorting of very fine sediment fractions and neglects the effects of phase lags of fine fractions. Therefore, a modification of this model based on experimental results was proposed.

_{i}< 0.20 mm and coarse fractions ${({q}_{c}^{+/-})}_{calc.}$ with diameter of d

_{i}≥ 0.20 mm, as follows:

_{i}< 0.2 mm) in the input mixture;

_{i}≥ 0.2 mm) in the input mixture;

_{i}< 0.2 mm), calculated using the Kaczmarek et al. 2022 model;

_{i}≥ 0.2 mm), calculated using the Kaczmarek et al. 2022 model;

_{f}—fraction with a diameter of d

_{i}= 0.2 mm.

_{i}< 0.2 mm) in the mixture collected from the trap;

_{i}≥ 0.2 mm) in the mixture collected from the trap.

#### 3.1.2. Grain Size Distributions of Transported Sediments

_{i}< 0.2 mm) and coarse fractions (d

_{i}≥ 0.2 mm):

_{i}< 0.2 mm) and coarse (d

_{i}< 0.2 mm) fractions for any given granulometric distribution of noncohesive sediments with the proportion of very fine and fine fractions, it is suggested to use the correction factors described by relations (11), (12), (13), and (14). Then the proportion of fine and very fine fractions ${({n}_{f}^{+})}_{st}$ and coarse fractions ${({n}_{c}^{+})}_{st}$ caught in the trap in the crest phase can be calculated as the transport ratio of fine/coarse fractions to the total transport in the crest phase, according to the following equations:

#### 3.2. Free Stream Velocity Measurements

#### 3.3. PIV Measurements

_{A}= −0.16 m and x

_{B}= 0.16 m. The instantaneous fluxes expressed by (24) were conditionally averaged over one wave period for consecutive waves. This conditional average refers to the negative (opposing the wave) and positive (following the wave) fluxes through the given cross section, which may be expressed as:

- 1.
- As would be expected, the flux ${\widehat{q}}_{B}$ within the boundary layer (thickness $\delta \approx 0.5\mathrm{cm}\approx {\delta}_{m})$ outgoing from the computational area and directed to the trap (positive values) reached higher values than the absolute values of the flux ${\widehat{q}}_{B}$ returning to the computational area (Table 6). However, the more very fine fractions in the bottom, the more the outgoing and returning fluxes ${\widehat{q}}_{B}$ balanced each other. This was understandable, because more very fine fractions in the fluxes ${q}_{f1}^{+}$, ${q}_{f2}^{+}$, and ${q}_{f3}^{+}$ meant a larger (after they were mixed) depth-averaged return flux. As a result, the balancing of the outgoing and returning fluxes ${\widehat{q}}_{B}$ occurred closer to the bottom, at the boundary of the near-wall layer (Figure 1). As a result, high above the bottom, the returning fluxes ${\widehat{q}}_{B}$ were larger than those outgoing the control area (Figure 1). This was especially visible in cases of stronger bottom impacts, which was confirmed by observations of ${\widehat{q}}_{B}$ (Table 6) in the area high above the near-wall layer ($\delta \approx 2.0\mathrm{cm}$), where the proportion of suspended fine and very fine fractions in this flux was dominant. It is worth noting that the above effect occurred in the crest phase, when it could be expected that the orbital velocities in the returning flux ${\widehat{q}}_{B}$ were smaller than in the outgoing flux ${\widehat{q}}_{B}$. In addition, the higher proportion of very fine fractions in the fluxes ${\widehat{q}}_{B}$ resulting from their elevation above the boundary layer may imply their absence from the bottom, thus causing an increase in the bottom roughness and a decrease in the flux ${q}_{st}^{+}$ entering the trap;

- 2.
- In the trough phase, the fluxes ${\widehat{q}}_{A}$ reached absolute values higher than fluxes ${\widehat{q}}_{B}$ in the crest phase (Table 6). The above observation seemed to be understandable due to the increased presence of very fine fractions in the computational area caused by the returning flux ${\widehat{q}}_{B}$. However, the increased differences between the absolute values of fluxes ${\widehat{q}}_{A}$; i.e., those outgoing from the computational area and directed to the trap (negative values) and those returning, clearly indicated a significant contribution ${q}_{st}^{-}$ in the flux ${\widehat{q}}_{A}$. This suggested a significant increase in ${q}_{st}^{-}$ by expanding the flux ${q}_{st1}^{-}$ at the expense of ${q}_{f3}^{-}$. On the other hand, the increased amount of very fine fractions in the trough phase implied an increase in the average postdepth concentration of these fractions, and in effect, by forming an envelope around coarse grains, they facilitated their transport. As a result, an increase in the amount of coarse fractions (d
_{i}≥ 0.20 mm) was to be expected in the flux ${q}_{st}^{-}$ entering the trap A. The above effects were so strong (Table 5) that even in areas located higher up the bottom ($\delta \approx 2.0\mathrm{cm}$), but under stronger conditions, the absolute values of the flux ${\widehat{q}}_{A};$ i.e., outgoing from the computational area and directed to the trap, remained larger than the returning flux ${\widehat{q}}_{A}$, as the latter was formed only by the fluxes ${q}_{f1}^{-}$ and ${q}_{f2}^{-}$.

#### 3.4. Correction Factors for Sediment Fluxes

_{2.5}; i.e., the dimensionless bottom friction calculated using the Kaczmarek et al. 2022 model for the maximum tangential stress during the wave period. The coefficients were calculated using Equations (11) ÷ (14) and then approximated using a correlation curve with a coefficient of determination ${R}^{2}\ge 0.80$. In order to obtain such a high value of fit, the few results of the calculated coefficients that deviated significantly from the correlation curve had to be omitted in some cases. The deviations were mainly in the case of sand with a content of very fine grains (type D; d

_{50}= 0.14). In such cases, the arithmetic mean values of the calculated values of ${\beta}_{1}^{+/-}$ and ${\beta}_{2}^{+/-}$ for all measurements of a given case were taken to calculate the modified transport ${q}_{st}^{+/-}$. These magnitudes are marked with triangles in Figure 8a–d. It is worth noting here that this behavior of the coefficients ${\beta}_{1}^{+/-}$ and ${\beta}_{2}^{+/-}$ for a bed composed of sediments with a large number of fine and very fine fractions was due to the fact that, as previously mentioned, an increase in these fractions caused an increase in fluxes ${q}_{f1}^{+/-}$,${q}_{f2}^{+/-}$, and ${q}_{f3}^{+/-}$ at the expense of ${q}_{st}^{+/-}$, which consequently caused a dramatic reduction of these coefficients. In turn, an increase in hydrodynamic actions on the bottom (increase in θ

_{2.5}) caused an increase in fluxes ${q}_{f1}^{+/-}$,${q}_{f2}^{+/-}$, and ${q}_{f3}^{+/-}$, as well as ${q}_{st}^{+/-}$, and this implied an increase in the coefficients ${\beta}_{1}^{+/-}$ and ${\beta}_{2}^{+/-}$ along with the increase in θ

_{2.5}.

#### 3.5. Calculated Sediment Transport versus Measurements

_{i}< 0.20 mm, then Eq. ${({q}_{c}^{+})}_{calc.}={({q}_{c}^{+})}_{st}$ (Figure 10b) would apply, while the total flux ${q}_{st}^{+}={({q}_{f}^{+})}_{st}+{({q}_{c}^{+})}_{st}$ would be reduced only by the magnitude of fluxes of fine fractions ${q}_{f3}^{+}$ [36] (cf. Kaczmarek et al., 2022). For the trough phase, in such a situation the relations ${({q}_{c}^{-})}_{st}={({q}_{c}^{-})}_{calc.}$ and ${q}_{st}^{-}={({q}^{-})}_{calc.}$ would be applied as the flux of fine fractions ${q}_{f3}^{-}$ flowed into the trap, feeding the total flux ${q}_{st}^{-}$.

#### 3.6. Calculated Grain Size Distributions versus Measurements

_{50}< 0.20 mm) and a very small amount of coarse fractions, only the calculated and measured total number of fine and very fine fractions ${n}_{f}^{+/-}={\displaystyle \sum}_{i=1}^{{N}_{f}}{n}_{fi}^{+/-}$ and the total amount of coarse fractions ${n}_{c}^{+/-}={\displaystyle \sum}_{i={N}_{f}}^{N}{n}_{ci}^{+/-}$ were used. Calculations of the granulometric compositions were also carried out with the Kaczmarek et al. 2022 model, using Equations (17) and (18). Calculated granulometric compositions were compared with measurements of granulometric compositions of sediments caught in traps A and B for the trough and crest phases, respectively. Additionally, Figure 12 shows the measurement results of the initial grain size distribution with the proportion of fine and very fine fractions ${n}_{fi}$ and coarse fractions ${n}_{ci}$ of sediments in the control area.

_{i}< 0.20 mm) and an underestimation of the content of coarse fractions (d

_{i}≥ 0.20 mm). This was due to the fact that the Kaczmarek et al. 2022 model did not take into account the effects of suspension of the finest fractions in the higher layer above the bottom and the effect of phase lags of the fine fractions, and as a result, the incomplete flux of fine and very fine fractions collected from the bottom in the crest phase fell entirely into trap B (Figure 12a,c,e,g). In the trough phase, the model of Kaczmarek et. al. 2022 did not take into account the effect of finest fractions on the increase in the coarse fraction content (Figure 12b,d,f,h) flowing into trap A. Therefore, it is worth emphasizing that the theoretical prediction, which took into account the correction coefficients resulting from the above effects, returned values similar to the measured ones.

## 4. Conclusions

- Total sediment transport with very fine and fine (d
_{i}< 0.20 mm) and coarse (d_{i}≥ 0.20 mm) fractions, in both the crest and trough, consisted of the following components:- ●
- Transport of outgoing sediment fluxes from the control area that were deposited in adjacent control areas in both directions. The movement of grains in these fluxes followed horizontal projection trajectories of different but equally probable lengths;
- ●
- Transport of fine sediment fluxes returning to the initial area in a suspended state. These fluxes were the result of phase lags between the water and sediment. Grains outgoing from the control area in the crest phase returned and were retained in this area. Then they flowed out in the trough as a flux that was deposited in the area adjacent to the trough side;
- ●
- Flux transports of very fine sediments returning to the initial area in the crest and trough phases in a suspended state—high above the bottom. These fluxes resulted from vertical sorting of the granulometrically heterogeneous sediment.

- An increase in the amount of very fine fractions in the return flow outgoing in the crest phase meant a loss in these fractions in the bottom, and thus an increase in the bottom roughness and grain movement resistance. As a result, the transport of coarse fractions outgoing in the flux from the control area and feeding the adjacent area decreased, and the transport of fine and very fine fractions decreased significantly due to the return of these fractions to the initial area. In the situation in which there were no very fine fractions in the bottom, only a loss in fine fractions was observed in the adjacent area due to the phase-lag effect.
- An increase in the amount of very fine fractions in the return flow outgoing in the trough phase caused an increase in the depth-averaged concentration of these fractions, and as a result, they formed an envelope around coarse grains, making the transport of these fractions easier, and significantly increasing the transport. In the case of a small amount of very fine fractions in the bottom, the above effect of these fractions on the transport of coarse grains was negligibly small, and the total flux outgoing in the trough was deposited in the adjacent area.
- The results of experimental investigations were compared with the results of theoretical analysis based on the three-layer model of Kaczmarek et al. 2022. As this model did not take into account the above-mentioned effects related to the presence of fine and very fine fractions, a modification of this model with the above-mentioned effects was proposed. Transport calculations were conducted separately for fluxes of very fine and fine fractions, coarse fractions, and total fractions outgoing in the crest and trough phase from the initial area and deposited in adjacent control areas. The consistency of the modified model with the measurements was achieved within plus/minus a factor of 2 of the measurements, as shown by plotted agreement lines. In addition, calculations of the granulometric distributions of sediment retained in the adjacent areas from the crest and trough were also carried out. Again, calculations were conducted separately for fine and very fine fractions, coarse fractions, and their sum. The calculated granulometric compositions were compared with measurements, and satisfactory agreement was obtained.
- Modification of the Kaczmarek et al. 2022 model was carried out based on four coefficients that corrected for fluxes of fine and very fine fractions and coarse fractions that fed adjacent control areas from the crest and trough. For sands with a relatively low content of fine fractions (d
_{50}≥ 0.20 mm), it was possible to find a functional relationship of these coefficients with a coefficient of determination R^{2}> 0.80. For sands with a dominant amount of fine and very fine fractions (d_{50}< 0.20 mm), such a relationship could not be obtained, suggesting the need for further experimental studies in this area. - An interpretation of the present laboratory experimental results in the natural situations could be easily made, as the free stream velocities were defined as the surface-wave-induced orbital velocity at the edge of the wave boundary layer, and the sediment transport was the result of the coexistence of wave motion asymmetry and compensatory return current. The present analysis of deposition from traps on the sides is highly recommended for engineering practices in coastal zone because the determination of sediment transport during the wave crest and wave trough is needed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Sediment particle trajectories during the wave crest ${q}_{st}^{+}$ and trough ${q}_{st}^{-}$ sediment transport.

**Figure 3.**Experimental setup in a wave flume for measurements of wave-induced sediment transport: (

**a**) side view of the wave flume with the sand box implemented; (

**b**) side view of the wave flume.

**Figure 6.**Grain size distribution of quartz sands used in the experiment A: d

_{50}= 0.38; B: d

_{50}= 0.32; C: d

_{50}= 0.25; D: d

_{50}= 0.14.

**Figure 8.**Plots of sediment flux correction factors: (

**a**) fine fractions, crest phase; (

**b**) coarse fractions, crest phase; (

**c**) fine fractions, trough phase; (

**d**) coarse fractions, trough phase.

**Figure 9.**Comparison of calculated and measured transport during (

**a**) the crest ${q}^{+}$ and (

**b**) the trough ${q}^{-}$.

**Figure 10.**Comparison of calculated and measured transport during the crest q

^{+}in (

**a**) fine fractions ${q}_{f}^{+}$ and (

**b**) coarse fractions ${q}_{c}^{+}$.

**Figure 11.**Comparison of calculated and measured transport during the trough q

^{-}in (

**a**) fine fractions ${q}_{f}^{-}$ and (

**b**) coarse fractions ${q}_{c}^{-}$.

**Figure 12.**Comparison of calculated and measured grain size distributions in fine fractions and coarse fractions: (

**a**) grain A—crest; (

**b**) grain A—trough, (

**c**) grain B—crest; (

**d**) grain B—trough, (

**e**) grain C—crest; (

**f**) grain C—trough, (

**g**) grain D—crest; (

**h**) grain D—trough.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Water depth | h_{0} | 0.28/0.36 | m |

Wave height | H_{w} | 0.12 | m |

Test duration | T_{w} | 10 | min |

Wave peak period | T_{p} | 3.0 | s |

Representative diameter of bottom-building sediment grains | d_{50} | A: 0.38 B: 0.32 C: 0.25 D: 0.14 | mm |

Sediment density | ρ_{s} | 2.62 | g/cm^{3} |

Liquid density | ρ_{w} | 1.00 | g/cm^{3} |

Sediment porosity | n_{p} | 0.4 | – |

Test No. | Sand | h | T | H |
---|---|---|---|---|

T0134 | A | 0.28 | 3.0 | 0.12 |

T0135 | B | 0.28 | 3.0 | 0.12 |

T0136 | C | 0.28 | 3.0 | 0.12 |

T0137 | D | 0.28 | 3.0 | 0.12 |

T0128 | A | 0.36 | 3.0 | 0.12 |

T0129 | B | 0.36 | 3.0 | 0.12 |

T0130 | C | 0.36 | 3.0 | 0.12 |

T0131 | D | 0.36 | 3.0 | 0.12 |

System Item/Method | Description/Parameters |
---|---|

Data Acquisition: | |

Light source | Dual-head Nd-YAG 532-nm laser, 2 × 50 mJ @ 50 Hz |

Camera | Imager Pro HS 500 double-frame CCD, 1280 × 1024 @ 520 fps |

Data Processing: | |

Vector calculation | Time series of single frames: cross-correlation; multipass with decreasing window size; iterations: - Initial step: 128 × 128 pixel window, single pass, 50% overlap - Final steps: 64 × 64 pixel window, two passes, 75% overlap |

Options: - Image correction; - High-accuracy mode for final passes; | |

Masking functions | Geometric mask |

Type of Sand | d_{90}/d_{50}/d_{10} |
---|---|

A. Coarse quartz sand | 0.58/0.38/0.24 |

B. Medium quartz sand | 0.48/0.32/0.20 |

C. Fine quartz sand | 0.38/0.25/0.16 |

D. Very fine quartz sand | 0.19/0.14/0.08 |

Test No. | 0.5*a0 | a1 | b1 | a2 | b2 | a3 | b3 | a4 | b4 | ADV |
---|---|---|---|---|---|---|---|---|---|---|

T = 3 H = 0.12 H = 0.28 | −0.0182 | 0.0259 | −0.1934 | −0.0414 | 0.0616 | −0.0338 | 0.0137 | −0.0011 | −0.0097 | A |

−0.0198 | 0.0509 | −0.1758 | −0.0641 | 0.0550 | 0.0342 | 0.0293 | 0.0017 | −0.0110 | B | |

T = 3 H = 0.12 H = 0.36 | −0.0219 | 0.0148 | −0.1876 | −0.0249 | 0.1035 | 0.0591 | −0.0387 | −0.0305 | −0.0105 | A |

−0.0241 | −0.0174 | −0.1824 | 0.0257 | 0.0858 | 0.0292 | −0.0602 | −0.0332 | 0.0098 | B |

δ = 0.5 cm | ||||
---|---|---|---|---|

Case | Trough | Crest | ||

${\widehat{\mathit{q}}}_{\mathit{A}}$ from Sand Trap | ${\widehat{\mathit{q}}}_{\mathit{A}}$ to Sand Trap | ${\widehat{\mathit{q}}}_{\mathit{B}}$ to Sand Trap | ${\widehat{\mathit{q}}}_{\mathit{B}}$ from Sand Trap | |

Sand A h = 0.36 m | 0.00012 (8.6%) | −0.00020 (4.5%) | 0.00009 (6.9%) | −0.00005 (6.3%) |

Sand B h = 0.36 m | 0.00018 (3.7%) | −0.00025 (5.6%) | 0.00009 (8.3%) | −0.00006 (9.6%) |

Sand C h = 0.36 m | 0.00022 (5.2%) | −0.00025 (3.9%) | 0.00007 (13.1%) | −0.00006 (7.1%) |

Sand A h = 0.28 m | 0.00017 (4.8%) | −0.00028 (6.8%) | 0.00014 (11.1%) | −0.00009 (6.8%) |

Sand B h = 0.28 m | 0.00010 (14.8%) | −0.00022 (8.9%) | 0.00008 (21.2%) | −0.00009 (8.7%) |

Sand C h = 0.28 m | 0.00020 (8.1%) | −0.00026 (5.7%) | 0.00006 (16.4%) | −0.00007 (5.3%) |

δ = 2.0 cm | ||||

Case | Trough | Crest | ||

${\widehat{q}}_{A}$
from sand trap | ${\widehat{q}}_{A}$
to sand trap | ${\widehat{q}}_{B}$
to sand trap | ${\widehat{q}}_{B}$
from sand trap | |

Sand A h = 0.36 m | 0.00067 (7.2%) | −0.00061 (7.0%) | 0.00016 (9.1%) | −0.00026 (7.3%) |

Sand B h = 0.36 m | 0.00094 (2.5%) | −0.00091 (8.1%) | 0.00020 (10.3%) | −0.00035 (11.2%) |

Sand C h = 0.36 m | 0.00105 (3.1%) | −0.00091 (5.2%) | 0.00021 (15.2%) | −0.00031 (8.5%) |

Sand A h = 0.28 m | 0.00102 (4.3%) | −0.00117 (7.6%) | 0.00042 (17.0%) | −0.00049 (8.6%) |

Sand B h = 0.28 m | 0.00065 (11.7%) | −0.00072 (10.2%) | 0.00024 (24.5%) | −0.00038 (11.4%) |

Sand C h = 0.28 m | 0.00102 (4.6%) | −0.00112 (7.6%) | 0.00016 (12.6%) | −0.00035 (6.4%) |

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## Share and Cite

**MDPI and ACS Style**

Radosz, I.; Zawisza, J.; Biegowski, J.; Paprota, M.; Majewski, D.; Kaczmarek, L.M.
An Experimental Study on Progressive and Reverse Fluxes of Sediments with Fine Fractions in Wave Motion. *Water* **2022**, *14*, 2397.
https://doi.org/10.3390/w14152397

**AMA Style**

Radosz I, Zawisza J, Biegowski J, Paprota M, Majewski D, Kaczmarek LM.
An Experimental Study on Progressive and Reverse Fluxes of Sediments with Fine Fractions in Wave Motion. *Water*. 2022; 14(15):2397.
https://doi.org/10.3390/w14152397

**Chicago/Turabian Style**

Radosz, Iwona, Jerzy Zawisza, Jarosław Biegowski, Maciej Paprota, Dawid Majewski, and Leszek M. Kaczmarek.
2022. "An Experimental Study on Progressive and Reverse Fluxes of Sediments with Fine Fractions in Wave Motion" *Water* 14, no. 15: 2397.
https://doi.org/10.3390/w14152397