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

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{i}diameters. This is most likely due to the complexity of physical processes involved in transport, which is difficult to describe using simple empirical formulas that prevailed at a time when computer calculations were not possible. With the development of computer technology and numerical methods, as well as the growing importance of hydraulic engineering, attempts have been made to develop theoretical descriptions that allow the prediction of granulometrically heterogeneous sediment transport.

_{i}< 0.20 mm, and does not take into account the effects related to the bed slope, such as the effect of gravitational forces on sediment grains and the additional effect of reduced or increased pressure when changing the flow cross-section, it was proposed to modify the theoretical model with the above-mentioned effects.

## 2. Materials and Methods

#### 2.1. Theoretical Investigations

#### 2.1.1. Description of the Flow

_{i}< 0.20 mm, while fluxes ${q}_{f2}^{+/-}$ and ${q}_{f3}^{+/-}$ are the result of vertical sorting of granulometrically heterogeneous sediment, resulting in very fine fractions being lifted very high above the bed. It is worth recalling that the return fluxes of suspended sediments ${q}_{f1}^{+/-}$, ${q}_{f2}^{+/-}$ and ${q}_{f3}^{+/-}$ mix with each other and form a depth-averaged flux during the return phase:

_{i}≥ 0.2 mm). Another effect related to the bed slope is that caused by pressure gradients. It consists in the formation of areas with reduced or increased pressure occurring at varying flow cross sections. In fact, when the movement of sediment over a sloping bed is considered, it is possible to see a decrease in the flow cross-section and a decrease in depth in the wave direction. Consequently, as the cross-section decreases in the crest phase, velocities increase and pressure decreases, and a negative pressure gradient is created. It is worth noting that sediment flow increases because the frictional forces that cause it increase with the additional forces from the negative pressure gradient. In the trough, the situation is reversed—frictional forces causing sediment movement are reduced by forces from a positive pressure gradient. When the cross-section increases and the pressure increases and a positive pressure gradient is generated then the velocities decrease and, as a result, the intensity of sediment transport also decreases. Therefore, it can be expected that the outgoing sediment flows toward the trap in the wave trough phase, as well as the returning flows, will be smaller than the flows in the flat bed situation. Especially in the zone near the bed, mainly coarse grains will be retained and will fall directly into the trap, while the return flux ${q}_{f3}^{-}$ will feed the flux ${q}_{st}^{-}$. It is expected that the above effect, will have less impact on the fluxes of suspended grains ${q}_{f1}^{+/-}$ and ${q}_{f2}^{+/-}$, but always the return fluxes will be smaller than the fluxes for a flat bed due to the smaller hydrodynamic forcing. In the wave crest phase, the situation is reversed. In this phase, it is expected that due to greater hydrodynamic forcing, the fluxes entering the trap will be larger than in the flat bed situation, which means that the return flow ${q}_{f3}^{+}$ will also feed the flux ${q}_{st}^{+}$. However, at the same time, the return fluxes will decrease, as more of the total flux leaving the control area will feed the trap, but still they are greater than the fluxes for a flat bed due to the greater hydrodynamic forcing.

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

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

#### 2.1.3. 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 non-cohesive sediments with the proportion of very fine and fine fractions over a sloping bed, it is proposed to use correction factors described by relations (8)–(11). Then the proportion of fine and very fine fractions ${\left({n}_{f}^{+}\right)}_{st}$ and coarse fractions ${\left({n}_{c}^{+}\right)}_{st}$ caught in the trap in the crest phase over the sloping bed can be calculated as the transport ratio of fine/coarse fractions to the total transport in the crest phase, according to the following formulas:

#### 2.2. Experimental Investigations

#### 2.2.1. Experimental Set-Up

#### 2.2.2. Scope of the Measurement

- 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 equal to sand A: = 6%, B: = 10.5%, C: = 26%, respectively).

## 3. Results and Discussion

#### 3.1. Free Stream Velocity Measurements

#### 3.2. PIV Measurements

_{A}= −0.16 m and x

_{B}= 0.16 m. The instantaneous fluxes expressed by (21) 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:

- As can be assumed, in the situation of a sloping bed, the flux ${\widehat{q}}_{B}$ inside the near-wall layer (thickness $\delta \approx 0.5\text{}\mathrm{cm}\approx {\delta}_{m}$, where ${\delta}_{m}$ is the thickness of the boundary layer as determined by the Kaczmarek et al. 2022 model) from the computational area and directed to the trap (positive values) reaches values clearly greater than the absolute values of flux ${\widehat{q}}_{B}$ returning to the computational area (Table 4). This is understandable, as the effect of a negative pressure gradient becomes apparent, which causes an increase in hydrodynamic forcing and an increase in flux values ${\widehat{q}}_{B}$. For a flat bed, in the lack of a negative pressure gradient, the magnitudes of fluxes leaving and returning to the computational area are much smaller. Under the conditions of a sloping bed, only in case of sand D, i.e., in the situation of presence of a very large number of fine fractions in the bed, the fluxes ${\widehat{q}}_{B}$ returning are larger than those leaving. This is because the large number of very fine fractions in the fluxes ${q}_{f1}^{+}$, ${q}_{f2}^{+}$ and $\text{}{q}_{f3}^{+}$ means a larger (after they are mixed) and closer to the bed averaged by depth return fluxes ${\widehat{q}}_{B}$, at the boundary of near-wall layer.In case of sediments A, B, C, the returning fluxes ${\widehat{q}}_{B}$ high above the bed ($\delta \approx 2.0\text{}\mathrm{cm}$) are balanced with the fluxes leaving the control area. Only in the case of sediment D, the returning fluxes ${\widehat{q}}_{B}$ are larger than the outgoing fluxes, as the proportion of suspended fine and very fine fractions in this flux is dominant. In case of a flat bed, the returning fluxes (Table 4) are slightly larger, while in the case of a sloping bed, the negative pressure gradient causes more material (including fluxes ${q}_{f3}^{+}$) falls into the trap. As a result, the return fluxes are slightly smaller.In summary, it can be expected that higher hydrodynamic forcing caused by a negative pressure gradient will cause the fluxes measured over a sloping bed to be closer to those calculated by the Kaczmarek et al. model of fluxes ${\left({q}_{st}^{+}\right)}_{calc.}$, than in the flat bed situation;
- Under sloping bed conditions, the fluxes ${\widehat{q}}_{A}$ both outgoing and entering the trap (with a minus sign) in the wave trough phase are smaller than the fluxes ${\widehat{q}}_{A}$ under flat bed conditions. This is due to weaker hydrodynamic conditions caused by the effect of a positive pressure gradient, which reduces transport. In addition, at the bed the flux ${q}_{f3}^{-}$ outgoing towards the trap in the area of positive pressure gradient additionally feeds the flux ${q}_{st}^{-}.\text{}$ As a result, the fluxes entering traps will be significantly larger than those outgoing.It is also worth noting, that in case of a sloping bed, the fluxes ${\widehat{q}}_{A}$ directed to traps (for sediments A, B, C) in the trough phase of the wave reach absolute values smaller than the fluxes ${\widehat{q}}_{B}$ in the crest phase of the wave (Table 4). The above observation seems to be understandable due to the effects occurring over the sloping bed, which increase fluxes ${\widehat{q}}_{B}$ and decreasing the fluxes ${\widehat{q}}_{A}$. In turn, the fluxes ${\widehat{q}}_{B}$ outgoing from the trap (with a minus sign) are similar in value to the fluxes ${\widehat{q}}_{A}$ entering the trap on the opposite side (with a minus sign). This represents the opposite situation to that on a flat bed (Table 4), where fluxes ${\widehat{q}}_{A}$ entering the trap are clearly larger than the fluxes ${\widehat{q}}_{B}\text{}$ outgoing on the opposite side from the trap. Furthermore, taking into account the fact that under sloping bed conditions the flux in the trough phase ${\widehat{q}}_{A}$ directed to the trap is smaller than the flux ${\widehat{q}}_{A}$ under flat bed conditions, it can be expected that the effect of fine fractions in the trough of the wave on the acceleration of coarse fractions will be much smaller. On the other hand, the increased amount of material flowing into the trap (including very fine fractions) in the wave crest leads to expect that the effect of flushing very fine fractions from the bed occurring in the case of a flat bed will not take place in the situation of a sloping bed. Finally, it is worth noting that the effects caused by positive pressure gradient are so strong (Table 4) that even in areas higher up the bed ($\delta \approx 2.0\text{}\mathrm{cm})\text{}$for all types of sediments, the absolute values of flux ${\widehat{q}}_{A}$ from the calculation area and directed to the trap remain greater than the flux ${\widehat{q}}_{A}$ returning. This is because the latter is formed only by the fluxes ${q}_{f1}^{-}$ and ${q}_{f2}^{-}$. This is the opposite situation from that on a flat bed, where the fluxes were balanced.

#### 3.3. Correction Factors for Sediment Fluxes

_{2.5}, i.e., dimensionless bed friction calculated by the Kaczmarek et al. 2022 model for the maximum tangential stress during the wave period. The coefficients were calculated using Formulas (8) ÷ (11) and then approximated by a correlation curve with a determination coefficient ${R}^{2}\ge 0.80$. In order to obtain such a high fit value, a few results of calculated coefficients significantly deviating from the correlation curve were omitted. The deviations concerned cases of sand with a high content of very fine grains (type D; d

_{50}= 0.14). In such cases, for the calculation of modified transport ${q}_{st}^{+/-}$ values were taken as arithmetic averages of the calculated quantities of ${\beta}_{1}^{+/-}$ i ${\beta}_{2}^{+/-}$ for all measurements of a given case. These quantities are marked with triangles in Figure 6a,b,c,d. Radical reduction of the coefficients ${\beta}_{1}^{+/-}$ (Figure 6a,c) for a bed composed of sediments with a large amount of fine and very fine fractions is due to the increase in fluxes ${q}_{f1}^{+/-}$,$\text{}{q}_{f2}^{+/-}$ and ${q}_{f3}^{+/-}$ at the expense of ${q}_{st}^{+/-}$. In addition, an increase in hydrodynamic impacts on the bed (increase in θ

_{2.5}) results in both an increase in fluxes ${q}_{f1}^{+/-}$, $\text{}{q}_{f2}^{+/-}$ and $\text{}{q}_{f3}^{+/-}$, as well as ${q}_{st}^{+/-}$, and this implies an increase in the coefficients ${\beta}_{1}^{+/-}$ with the increase of θ

_{2.5}. In turn, a dramatic increase in the coefficients ${\beta}_{2}^{+/-}$ for sand D implies an increased amount of fine and very fine fractions in the flow and their significant effect on increasing the proportion of coarse fractions in the transport (Figure 6b,d).

#### 3.4. Comparison of Transport Calculations with Mesurement Results

#### 3.5. Comparison of Grain Size Distributions Calculations with Mesurement Results

_{50}< 0.20 mm) for sand D with a large amount of fine and very fine fractions ${n}_{f}^{+/-}={{\displaystyle \sum}}_{i=1}^{{N}_{f}}{n}_{fi}^{+/-}$ and the summed amount of coarse fractions ${n}_{c}^{+/-}={{\displaystyle \sum}}_{i={N}_{f}}^{N}{n}_{ci}^{+/-}$. The proportion of fine and very fine fractions ${({n}_{f}^{+/-})}_{st}$ and coarse ${({n}_{c}^{+/-})}_{st}$ was calculated by Formulas (16) and (17) for the crest phase and by Formulas (18) and (19) for the trough phase. Granulometric compositions were also calculated using the Kaczmarek et al. 2022 model, using Formulas (14) and (15). The calculated granulometric compositions were compared with measurements of the granulometric compositions of sediments caught in trap A and B, respectively, in the trough and wave crest phases above the sloping bed. In addition, Figure 13 shows the measurement results of initial grain size distribution, with the proportion of fine and very fine fractions ${n}_{fi}$ and coarse fractions ${n}_{ci},$ sediments in the control area.

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

_{i}≥ 0.20 mm). This is due to the fact that the Kaczmarek et al. 2022 model does not take into account the effects of suspending the finest fractions in the higher layer above the bed and their return in the trough phase, and the effect of phase lags of fine fractions in the crest phase, and as a result, the incomplete flux of fine and very fine taken from the bed in the crest phase falls entirely into trap B (Figure 13a,c,e,g). In the trough phase, the Kaczmarek et. al. 2022 model does not take into account the effect of finest fractions on the increase in content of coarse fractions (Figure 13b,d,f,h) falling into trap A. Therefore, it is worth emphasizing that the theoretical prediction with correction factors, resulting from the above effects for the sloping bed, gives values close to the measured ones.

## 4. Conclusions

- The vertical structure of total sediment transport with the content of very fine and fine (d
_{i}< 0.20 mm) and coarse (d_{i}≥ 0.20 mm) fractions, both in the crest and trough of the wave, consists of components:- transport of outgoing sediment fluxes from the control area, which are deposited in adjacent control areas in both directions;
- transport of fine and very fine sediment fluxes returning to the initial area in a suspended state;

- Experimental results were compared with the results of theoretical analysis, based on the three-layer model of Kaczmarek et al. 2022. 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, as well as does not take into account the effects related to the bed slope, i.e., effect of gravitational forces acting on grains, and the effect of pressure gradients when changing the flow cross section. Therefore, it was proposed to modify this model with the above-mentioned effects. Subsequently, calculations of the transport of fluxes of very fine and fine fractions, coarse fractions and total fractions outgoing in the crest and trough phases from the initial area and deposited in adjacent control areas were carried out and compared with the measurement results. The agreement of calculation results with measurements within +/− a factor two of the measurements was obtained. Calculations of granulometric distributions of sediments retained in adjacent areas from the crest and trough of the wave were also carried out. The calculated granulometric compositions were compared with the measurements and satisfactory agreement of the results was obtained for fine, very fine and coarse fractions;
- Modification of the Kaczmarek et al. 2022 model was carried out based on four coefficients that correct the fluxes of fine and very fine fractions and coarse fractions that feed adjacent control areas on the crest and trough side of the wave. For sands with a relatively low content of fine fractions with d
_{50}≥ 0.20 mm, it was possible to find a functional relationship of these coefficients with a determination coefficient 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 research in this area. - Sediment transport over the sloping bed in the wave crest increases compared to the flat bed, due to the zone of negative pressure gradient. In the wave crest phase, due to higher hydrodynamic forcing, the fluxes deposited in the adjacent control area (falling into the trap) are larger than those in the flat bed situation. This results in better agreement between the results of calculations by the Kaczmarek et al. 2022 model and the results of measurements compared to those from a flat bed. This agreement deteriorates in case of sediments with a high content of fine and very fine grains, when the fluxes of coarse fractions falling into the trap and the fluxes of fine fractions returning to the initial area are increased;
- In the trough, the situation is reversed—the intensity of sediment transport decreases due to the zone of positive pressure gradient. Due to lower hydrodynamic forcing, the outgoing fluxes toward the adjacent area, as well as the return fluxes in the wave crest are smaller than the fluxes in a flat bed situation. Virtually all of the outgoing flux feeds into the adjacent area (trap), and only a relatively small flux of the finest fractions returns in the wave crest to the initial area. This results in almost perfect agreement between calculations by the Kaczmarek et al. 2022 model and measurements. However, in case of sediment with a large number of fine and very fine fractions, their increased amount in suspension has a decisive influence on the transport of coarse fractions and finally results in an underestimation of the measured transport of coarse fractions by the Kaczmarek et al. 2022 model;
- The influence of gravity forces reveals itself very strongly only in the trough phase when sediment is composed of coarse fractions and a small amount of fine and very fine fractions. However, in case of the greater amount of fine and very fine fractions in the bed, they have a decisive influence on the transport of coarse fractions in the trough phase, far exceeding the influence of gravity forces.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Vertical structure of the wave-induced progressive and reverse sediment fluxes over slopped bed.

**Figure 2.**Experimental setup in a wave flume for measurements of wave-induced sediment transport on a sloping bed: (

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

**b**) top view of the wave flume with position of gauges and sand traps.

**Figure 3.**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 4.**Characteristic records of the along flume particle velocities recorded by ADV A (

**left**) and ADV B (

**right**).

**Figure 5.**Characteristic records of horizontal particle velocities recorded by ADV A (left) and ADV B (right).

**Figure 6.**Plots of sediment flux correction factors over a sloping seabed: (

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

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

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

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

**Figure 7.**Comparison of calculation and measurements results ${q}_{f}^{+}$ between flat (

**a**) and sloped bed (

**b**) cases during the crest.

**Figure 8.**Comparison of calculation and measurements results ${q}_{c}^{+}$ between flat (

**a**) and sloped bed (

**b**) cases during the crest.

**Figure 9.**Comparison of calculation and measurements results ${q}^{+}$ between flat (

**a**) and sloped bed (

**b**) cases during the crest.

**Figure 10.**Comparison of calculation and measurements results ${q}_{f}^{-}$ between flat (

**a**) and sloped bed (

**b**) cases during the trough.

**Figure 11.**Comparison of calculation and measurements results ${q}_{c}^{-}$ between flat (

**a**) and sloped bed (

**b**) cases during the trough.

**Figure 12.**Comparison of calculation and measurements results ${q}^{-}$ between flat (

**a**) and sloped bed (

**b**) cases during the trough.

**Figure 13.**Comparison of calculated and measured grain size distributions in fine fractions and coarse fractions in sloped bed case: (

**a**) sand A—crest; (

**b**) sand A—trough; (

**c**) sand B—crest; (

**d**) sand B—trough; (

**e**) sand C—crest; (

**f**) sand C—trough; (

**g**) sand D—crest; (

**h**) sand D—trough.

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

Water depth | h_{0} | 0.50 | 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 | – |

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 |

**Table 3.**Summary of Fourier series coefficients used as an input to numerical model for wave parameters: T = 3.0, Hw = 0.12.

0.5*a0 | a1 | b1 | a2 | b2 | a3 | b3 | a4 | b4 | ADV |
---|---|---|---|---|---|---|---|---|---|

−0.0216 | 0.0389 | −0.2504 | −0.0216 | 0.0358 | −0.0022 | 0.0216 | 0.0025 | 0.0107 | A |

−0.0395 | 0.0396 | −0.3023 | −0.0121 | 0.0611 | 0.0003 | 0.0198 | 0.0045 | 0.0116 | B |

**Table 4.**The comparison of the wave-average values of the phase-averaged fluxes ${\widehat{q}}_{\left(A/B\right)}$ in the flat bed conditions with the results in the sloped bed conditions.

δ = 0.5 cm—Flat Bed | ||||

Case | Trough h = 0.36 cm | Crest h = 0.28 cm | ||

${\widehat{\mathit{q}}}_{\mathit{A}}$ 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 | 0.00012 | −0.00020 | 0.00014 | −0.00009 |

(8.6%) | (4.5%) | (11.1%) | (6.8%) | |

Sand B | 0.00018 | −0.00025 | 0.00008 | −0.00009 |

(3.7%) | (5.6%) | (21.2%) | (8.7%) | |

Sand C | 0.00022 | −0.00025 | 0.00006 | −0.00007 |

(5.2%) | (3.9%) | (16.4%) | (5.3%) | |

δ = 2.0 cm—Flat Bed | ||||

Case | Trough h = 0.36 cm | Crest h = 0.28 cm | ||

${\widehat{\mathit{q}}}_{\mathit{A}}$ 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 | 0.00067 | −0.00061 | 0.00042 | −0.00049 |

(7.2%) | (7.0%) | (17.0%) | (8.6%) | |

Sand B | 0.00094 | −0.00091 | 0.00024 | −0.00038 |

(2.5%) | (8.1%) | (24.5%) | (11.4%) | |

Sand C | 0.00105 | −0.00091 | 0.00016 | −0.00035 |

(3.1%) | (5.2%) | (12.6%) | (6.4%) | |

δ = 0.5 cm—Sloped Bed | ||||

Case | Trough h = 0.36 cm | Crest h = 0.28 cm | ||

${\widehat{\mathit{q}}}_{\mathit{A}}$ 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 | 0.00004 | −0.00018 | 0.00033 | −0.00017 |

(8.0%) | (5.9%) | (4.0%) | (4.9%) | |

Sand B | 0.00004 | −0.00021 | 0.00039 | −0.00015 |

(7.7%) | (7.0%) | (3.5%) | (8.4%) | |

Sand C | 0.00004 | −0.00017 | 0.00033 | −0.00015 |

(7.5%) | (8.3%) | (4.2%) | (8.4%) | |

Sand D | 0.00005 | −0.00015 | 0.00010 | −0.00013 |

(6.3%) | (4.5%) | (9.6%) | (5.5%) | |

δ = 2.0 cm—Sloped Bed | ||||

Case | Trough h = 0.36 cm | Crest h = 0.28 cm | ||

${\widehat{\mathit{q}}}_{\mathit{A}}$ 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 | 0.00049 | −0.00044 | 0.00075 | −0.00078 |

(7.5%) | (8.4%) | (6.2%) | (4.3%) | |

Sand B | 0.00068 | −0.00085 | 0.00106 | −0.00098 |

(3.8%) | (8.7%) | (5.2%) | (7.0%) | |

Sand C | 0.00061 | −0.00076 | 0.00107 | −0.00094 |

(4.2%) | (8.8%) | (6.2%) | (7.0%) | |

Sand D | 0.00037 | −0.00073 | 0.00049 | −0.00085 |

(6.1%) | (5.3%) | (8.3%) | (4.3%) |

**Table 5.**Comparison of the correction factors over a sloping and flat seabed in the wave crest phase and wave trough phase.

Crest | ||||

Type of Sand | ${\mathit{\beta}}_{1}^{+}$—Sloped Bed Case | ${\mathit{\beta}}_{1}^{+}$—Flat Bed Case | ${\mathit{\beta}}_{2}^{+}$—Sloped Bed Case | ${\mathit{\beta}}_{2}^{+}$—Flat Bed Case |

Sand A | 0.278185 | 0.204274 | 0.96722 | 0.600155 |

Sand B | 0.374075 | 0.308164 | 0.95137 | 0.759586 |

Sand C | 0.535349 | 0.482891 | 0.924714 | 1.027727 |

Sand D | 0.258206 | 1.049027 | 5.387826 | 1.896533 |

Trough | ||||

Type of Sand | ${\mathit{\beta}}_{1}^{-}$—Sloped Bed Case | ${\mathit{\beta}}_{1}^{-}$—Flat Bed Case | ${\mathit{\beta}}_{2}^{-}$—Sloped Bed Case | ${\mathit{\beta}}_{2}^{-}$—Flat Bed Case |

Sand A | −0.33714 | −0.31215 | 0.278621 | 0.153719 |

Sand B | −0.22297 | −0.12580 | 0.320369 | 0.531003 |

Sand C | −0.03117 | 0.187276 | 0.390508 | 1.164859 |

Sand D | −0.21587 | 1.199482 | 6.076916 | 3.214157 |

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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 the Wave Motion over Sloped Bed. *Water* **2023**, *15*, 125.
https://doi.org/10.3390/w15010125

**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 the Wave Motion over Sloped Bed. *Water*. 2023; 15(1):125.
https://doi.org/10.3390/w15010125

**Chicago/Turabian Style**

Radosz, Iwona, Jerzy Zawisza, Jarosław Biegowski, Maciej Paprota, Dawid Majewski, and Leszek M. Kaczmarek.
2023. "An Experimental Study on Progressive and Reverse Fluxes of Sediments with Fine Fractions in the Wave Motion over Sloped Bed" *Water* 15, no. 1: 125.
https://doi.org/10.3390/w15010125