Formula for Sediment Transport Subject to Vertical Flows

Sediment transport is a geophysical phenomenon in which sediment particles are driven to move in streamwise and vertical directions by various forces. Almost all existing formulas of sediment transport were derived without considering vertical flows V, resulting in a large discrepancy between measured and predicted transport rates, as has been reported in the literature. This paper investigates the effect of vertical motion on sediment transport. It was found that upward fluid velocity increases particles' mobility, and downward motion increases particles stability. Furthermore, the investigation showed that decelerating flows can promote upward flow and vice versa. New equations were developed to express the influence of vertical motion on sediment transport. A reasonably good agreement between measured and predicted sediment transport rates was achieved. Disciplines Engineering | Science and Technology Studies Publication Details Yang, S. (2019). Formula for Sediment Transport Subject to Vertical Flows. Journal of Hydraulic Engineering, 145 (5), 04019013-1-04019013-11. This journal article is available at Research Online: https://ro.uow.edu.au/eispapers1/3755


Introduction
Sediment transport is a result of driving and resistance forces acting on particles.Some researchers believe that the boundary shear stress τ ð¼ρghSÞ alone can fully express the driving force of sediment transport.Thus, the equations of sediment transport by Einstein (1942), Meyer-Peter and Muller (1948), Yalin (1977), Engelund and Hansen (1972), and Ackers and White (1973) can be written as Φ ¼ fðτ Ã Þ, where Φ is the dimensionless form of sediment discharge g t , and τ Ã is the Shields (1936) shear stress parameter f¼τ =½ðρ s − ρÞgdg.
Because formulas that use τ Ã do not always provide good agreement with measured g t , some researchers have made attempts to use other parameters to express the driving force, like Velikanov's (1954) parameter, U 3 =ðghωÞ.This parameter has been widely used in Russia and China to express sediment transport.
As both the boundary shear stress τ and mean velocity U sometimes correlate poorly with measured sediment transport rate, another parameter, known as the stream power (¼τ U) was proposed 46 by Bagnold (1966), who hypothesized that the work used to trans-47 port sediment comes from the stream power.However Yalin (1977) 48 expressed concern that the stream power can be rewritten as ρgSq, 49 where q ¼ Uh ¼ Q=b, for a river reach where the channel slope S 50 is constant.The concept of stream power implies that the sediment 51 discharge is proportional to the discharge only.This is not correct, It was probably van Rijn (1984) who was the first one to make 65 attempts to include the influence of bedforms in the formulas for 66 sediment transport.By introducing a new parameter, the shear 67 velocity related grains u Ã 0 , he developed 7 equations that depend only 68 on two parameters, that is, T and d Ã , which are expressed as 69 where   (2) the mean velocity U or its dimensionless form U 3 =ðghωÞ, US=ω, T, and; (3) the stream power τ U.It is important to note that these three often give vastly different predictions.For example, in a simple laboratory experiment in which the flume slope, channel width, particle size (d 50 or ω), and discharge and other parameters are kept constant, only the bed roughness is changed.The formulas using U predict that the small velocity hindered by higher bed roughness will reduce g t .The formulas using τ predict that, at higher roughness, the higher τ will yield higher g t , because τ ¼ ρghS and higher h ¼ q=U are caused by the lower velocity, or else g t will be infinite if the roughness is so high that the velocity approaches zero.However, the models of stream power τ U state that 9 the variation of roughness will not change g t at all, because q = constant 10 , as pointed out by Yalin (1977).Therefore, the three models have totally different predictions.
In practice, the concept of sustainable development has been widely accepted worldwide, and many cities, such as Singapore, and Seoul, South Korea, want to replace their existing concrete drainage systems with natural plantation.City planners, water engineers, and citizens are keen to know the consequences of these replacements for sediment transport in a given return-period flood.
Obviously, no satisfactory answer can be obtained from the existing models to guide such projects, in which roughness is increased by vegetation.This highlights the fact that further investigation on sediment transport is needed even for steady and uniform flows.
Based on the fact that sediment transport is a near-boundary phenomenon 11 12 , Yang (2005) modified Bagnold's expression by using near-boundary parameters, that is, boundary shear stress τ , and near boundary velocity, which can be represented by u 0 Ã ; the energy used to support sediment transport is redefined as E ¼ τ u 0 Ã , and the average sediment velocity is assumed to be proportional to u 0 Ã rather than U as in Bagnold's expression.The following equation was obtained to express the total load of sediment transport: where k = a constant (=12.2),which is insensitive to other hydraulic parameters like Froude number, Reynolds number, relative roughness, and Rouse's number 13 (Yang et al. 2007); and E c (¼ρu 3 Ãc ) = critical energy needed to transport sediment.The arrows in Eq. (4) indicate that sediment transport is a vector, and its direction follows the near-bed velocity u 0 Ã , which is useful especially when the flow directions of upper and lower water layers are different, as in cases like large reservoirs, estuaries, and open seas.
Eq. ( 4) produces the highest correlation coefficient among the existing parameters, and on average it is 11% higher than that of the unit stream power (Yang 2005).All flow parameters used in Eq. ( 4) represent the characteristics of the boundary region (e.g., boundary shear stress, near bed velocity, and so forth), in which the majority of sediment particles are transported.
This simple review reveals that all models of sediment transport rely only on streamwise parameters, such as U, u 0 Ã , τ , E, and US.
All equations, including Eq. (4), predict that the higher the streamwise parameters are, the more particles will be transported (Liu and Chiew 2012).These models developed from unidirectional flows have been widely extended to complex cases, like unsteady, nonuniform flows, and large discrepancies have been observed.shear stress τ Ã in the following way: water can be determined by the following force balance equation: its density from ρ s to ρ 0 s by assuming that the particle size remains 212 unchanged, and the force balance equation is similar to Eq. ( 6): 213 From Eqs. ( 6) and ( 7), one can derive the following relationship: where u and v = streamwise and vertical time-averaged velocities in the x-and y-directions, respectively, as shown in Fig. 1.The vertical velocity can be determined from Eq. ( 9) as follows: In Eq. ( 10), ∂u=∂x is the gradient of streamwise velocity in the x-direction.The value is positive if the velocity becomes higher in the streamwise direction (accelerates) and negative if the flow is decelerating.Hence, an accelerating flow yields a negative or downward v, and a decelerating flow generates an upward or positive v.
At the permeable boundary, the fluid velocity must meet the Almost all equations of sediment transport were developed from steady and uniform flows in which the vertical velocity v is equal to 0, as stated in Eq. ( 10), and then were extended to flows with v ≠ 0. For example, the Shields diagram has been extended to express sediment movement in nonuniform/unsteady flows, and large discrepancies have been found.Eqs. ( 8) and ( 10) may provide a useful tool to improve the Shields diagram's application when its density is modified: where τ 0 c = critical shear stress with vertical velocity.Eq. ( 11) predicts that if the Shields number remains unchanged 21 , the observed critical shear stress in a decelerating flow (or phase) should be smaller than the critical shear stress without V s , and an accelerating flow (or phase) needs a higher shear stress to drive particles in motion.This can be seen by inserting Eq. ( 8) into Eq.( 11): Using Eq. ( 5), Eq. ( 12) can be rewritten as Eqs. ( 12) and ( 13) express the relationship between the modified Shields number τ 0 Ã and the original Shields number τ Ã .They predict that the original Shields number may significantly deviate from the Shields curve if there exists a vertical velocity V s .
Eq. ( 11) includes the influence of the vertical velocity.It demonstrates that an upward velocity reduces the apparent particle density 22 , thereby reducing the critical shear stress, whereas a downward 279 where 294 Differing from Eq. ( 14), Cheng and Chiew (1999) expressed 295 their data using the following empirical method: where they introduced a new parameter, V sc for a quick state; and 297 m ¼ 1 ∼ 2 depending on the characteristics of the sediments.The 298 value of V sc was determined by 299 where K = hydraulic conductivity; and λ = bed porosity.
300 By comparing Eqs. ( 14) and ( 16), one can see that the equations 301 are functionally similar to each other, but Eq. ( 14) is simpler with- Comparing Eqs. ( 14), (16), and (18), one can find that the conditions for τ 0 c ¼ 0 are, respectively, V s ¼ ω, V s ¼ V sc , and From the physical interpretation, it is apparent that Eq. ( 14) gives a reasonable limit.Eq. ( 16) may be not correct, because the calculated V sc could be less than or greater than ω.If V sc > ω, it implies that streamwise fluid force is still needed to initiate the particle movement even if all particles are in a suspended state, which is physically impossible.Similarly, if V sc < ω, it indicates that the streamwise force could be zero to move the particles when particles are not in the suspended mode 23 , which is also impossible.In addition, Eqs. ( 17) and (19) give different results for τ 0 c ¼ 0.

Vertical Velocity V s Induced by Nonuniformity
The vertical velocity in the sediment layer can be also induced by variations of water depth in space (steady nonuniform flow or dh=dx ≠ 0) as noted by Francalanci et al. (2008).This vertical velocity on the free surface can be seen from Eq. ( 10): where the subscript h stands for the free surface at y ¼ h.Using Leibniz's rule, one has For steady and nonuniform flows, if there is no seepage from the underground water, then Otherwise In natural conditions, groundwater and river water are exchanged.Thus, V s is always nonzero; that is, Y ≠ 0, which causes a discrepancy when the Shields curve is compared with field data.Fig. 3  In Fig. 3, the parallel lines represent Shields curve with different Y values from Eq. ( 13), it can be seen that the data points below/ above the original Shields curve (Y ¼ 0) can be explained by the nonzero Y, which explains why the measured critical shear stress significantly deviates from Shields' prediction.

Total Load of Sediment Transport Subject to V s
As mentioned previously, sediment transport is a joint effect of streamwise and vertical motions.The near bed streamwise motion can be represented by the parameters u 0 Ã and τ , and the vertical motion can be included in the apparent density ρ 0 s .Therefore, Eq. ( 4) can be modified into For sediment transport in a unidirectional flow, Eq. ( 24) becomes (Yang 2005) Eq. ( 25) may interpret the parameter T in Eq. ( 1), which was empirically discovered by van Rijn, but can be derived from Bagnold's modified theorem.Inserting Eq. ( 8) into Eq.( 25), one has Eq. ( 26) as 382 From Eqs. ( 26) and ( 27), one has Eq. ( 14) by changing Y.The following procedure was adopted in this study to measure the transport rate for a given injection rate: 1. Place the sand on the bed.Level the sand surface in the seepage region to the adjacent Perspex bed level.This is particularly supported by the "tearing bottom" phenomenon in the Yellow River in China (Chien and Wan 1998), in which large pieces of the riverbed, as much as several meters long, can be lifted to the free surface like a carpet.Observations show that this phenomenon always occurs when (1) hyperconcentrated flows are formed-that is, the settling velocity of the carpet, ω, is 26 reduced as the density of hyperconcentrated fluid could be as high as ρ ¼ 1,400 kg=m 3 (Chien and Wan 1998); (2) the water level is fallingthat is, a decelerating phase generates an upward velocity; and (3) the river flow is spatially decelerating from a narrow channel to a wider channel, which never occurs in channels with constant width or in a transitional reach from a wide channel to a narrow channel.Obviously, these facts imply that a higher and positive Y

Suspended Sediment Concentration Subject to V
The governing equation for suspended concentration can be derived from the mass continuity equation in the following form (Yang 2007): where C 0 = fluctuation of sediment concentration; C = timeaveraged sediment concentration by volume; V and W = vertical and lateral time-averaged velocities, respectively; and v 0 and w 0 are the velocity fluctuations in the y-and z-directions, respectively.
If the lateral gradient of Eq. ( 29) is negligible, the integration of Eq. ( 29) with respect to y yields the following equation: where the upper boundary condition at y ¼ h is applied to determine the integration constant.The importance of V for suspended sediment was noticed by researchers like Hawksley (1951), Fu et al.
The second term of Eq. ( 30) can be expressed by where ε s = sediment diffusion coefficient.Inserting Eq. ( 31) into Eq. ( 30), one obtains 507 Rouse assumes that ε s is proportional to the turbulent eddy 508 viscosity, that is 509 By inserting Eq. ( 33) into Eq.( 32), one obtains the modified 510 Rouse's law  The significance of Eq. ( 36) is that the net falling velocity of particles in rivers and coastal waters is not a constant but rather is variable.More sediment particles will "float," that is, higher concentrations appear V s has little effect as compared to the effect on small particles.This is why ripples can be observed only when a bed is made of fine particles like clay or sand, not of larger particles.
This also is a theoretical basis for riprap protection against local scours using large stones, that is, Y ≈ 0 in torrent flows.
As local scour holes are always related to upward velocity, it can be inferred that a bed dominated by downward velocity should have fewer scour holes or a relatively smooth surface, as shown in Fig. 6, in which an accelerating flow dominates the surface formation of the swash zone and forms a very smooth beach.Likewise, it can be 599 inferred that a riverbed on the concave side of a curved channel will 600 be very smooth, but scour holes will be found in its convex side due 601 to the action of secondary currents.

602
This study provides a theoretical framework for sediment 603 transport in unsteady and nonuniform flows and extends existing 604 formulas to complex flows, because the continuity equation, that 605 is, Eq. ( 9), is universal and can be applied in unsteady flows.Note: "Higher" and "lower" are in comparison to the "Region" column.a Observed by Nezu and Nakagawa (1993).67.A check of online databases revealed a possible error in this reference.The volume has been changed from '47' to '370'.Please confirm this is correct.A check of online databases revealed a possible error in this reference.The issue has been changed from '2' to '1-4'.Please confirm this is correct.A check of online databases revealed a possible error in this reference.The lpage has been changed from '184' to '190'.Please confirm this is correct.
68. Please provide the publisher or sponsor name and location (not the conference location) for Yang (2013).
69.This reference Yang and Tan (2008) is not mentioned anywhere in the text.ASCE style requires that entries in the References list must be cited at least once within the paper.Please indicate a place in the text, tables, or figures where we may insert a citation or indicate if the entry should be deleted from the References list.
phenomenon in natural waterways.The process of sediment transport is very complex and has attracted intensive research.Generally, sediment transport in wellcontrolled laboratory conditions can be reasonably predicted by many formulas available in the literature, but this is not true for field data.In these formulas, the basic parameters used are (Yang and Lim 2003) discharge Q, mean velocity U, channel slope S, water depth h, channel width b, particle size d (or settling velocity ω) and its gradation, sediment density ρ s , and gravitational acceleration g.Other fluid parameters include fluid density ρ and fluid viscosity ν.Parameters that reflect vertical motion, such as seepage velocity 5 or vertical pressure force induced by unsteadiness and nonuniformity, are totally neglected in almost all equations for sediment transport, even when the equations are extended to these complex cases.
52 because it excludes the influence of other hydraulic parameters such 53 as bedform roughness.In other words, if the flowrate is constant 54 along a river from upstream to downstream, the concept of stream 55 power indicates that the rate of sediment transport only depends on 56 the channel slope S. According to Yalin, this is unacceptable.57 Like the product of U and τ , the product of U and S 6 has been 58 tested against measured sediment transport.Yang (1973) empiri-59 cally found that, among the existing hydraulic parameters, the 60 parameter of unit stream power US=ω yielded the highest correla-61 tion coefficient with measured sediment concentration.This find-62 ing significantly advanced the knowledge of sediment transport and 63 improved the accuracy of sediment prediction.64 's model, the shear velocity u 0 Ã was introduced to 71 replace the mean velocity U.He argued that u 0 Ã is simple and con-72 veniently eliminates 8 the effect of bedform roughness.In his model, 73 the parameter of boundary shear stress was not included, because 74 "the energy gradient S is not an appropriate parameter for morpho-75 logical computations."76 In the literature, there are three different hydraulic param-77 eters used to express the driving force of sediment transport.

185 17 (
They argued that the numerator denotes the streamwise friction 186 force and the denominator represents the vertical force.Thus, if 187 pressure is not constant with respect to time and space, then the 188 Shields number must be modified.They used the pressure force 189 to modify the fluid density in the Shields number; high pressure 190 corresponds to a higher value of ρ and lower pressure to lower ρ.191 As shown in Fig. 1, a simpler model, similar to Francalanci 192 et al.'s 16 (2008) treatment, is considered in this study, in which ver-193 tical velocity, rather than the pressure force, represents vertical mo-194 tion, because velocity is more straightforward and convenient than 195 other parameters like pressure force or hydraulic gradient in the A permeable bed is represented by uniform spheri-197 cal particles with diameter d where the vertical velocity V s exists at 198 the interface.The value of the vertical velocity V s is much less than 199 U, but the presence of V s is very important in controlling sediment 200 transport because it alters the velocity distribution (Schlichting 201 1979), Reynolds shear stress distribution, flow resistance, and so 202 forth (Yang 2009a, b, c).Its influence on mass transport also needs 203 to be spelled out clearly Yang 2013; Alfadhli et al. 2014).204 For the particles shown in Fig. 1, the settling velocity ω in still 205 206 where C d = drag coefficient, which depends on the Reynolds 207 number R (¼ωd=ν) if R > 1; 000 and C d ¼ 0.45.208 In an environment in which the ambient fluid moves upward 209 with velocity V s , the net settling velocity is reduced to ω − V s .210 A reduction in the settling velocity could be treated by altering 211 214 where α ¼ C d 0=C d and α ¼ 1 are assumed in order to simplify the 215 mathematical treatment.216 Contrary to Francalanci et al.'s (2008) treatment, in which the 217 fluid density was modified in order to express the pressure influ-218 ence, Eq. (8) introduces the apparent particle density of ρ 0 s and 219 implies that the effects of nonhydrostatic pressure could be equiv-220 alently expressed by the variation of sediment density.If an upward 221 velocity (V s > 0) is present, Eq. (8) gives ρ 0 s < ρ s , and the sediment 222 can be represented as a lightweight material; a downward velocity 223 (V s < 0) increases ρ 0 s , implying that the sediment is more difficult to 224 move.Therefore, Eq. (8) demonstrates that an upward velocity V s 225 promotes the mobility of particles as they become "lighter," and a 226 downward velocity V s promotes the stability of particles as the 227 sediment becomes "heavier."228 For unsteady or nonuniform flows, the two-dimensional (2D) 229 continuity equation is

Fig. 1 .
Schematic diagram showing interaction of streamwise and F1apparent density, and the required critical 276 shear stress is higher.If the cases with and without vertical velocity 277 are compared, the critical shear stresses τ c and τ 0 c have the follow-278 ing relationship: 280where the correction coefficient β is introduced to express the re-281 lationship between the measured V and the vertical flow in the sedi-282 ment layer, V s , as shown in Fig.1.283 A comparison of Eq. (14) with experimental data by Cheng and 284 Chiew (1999), Kavcar and Wright (2009), and Liu and Chiew 285 (2012) is shown in Fig. 2. In Cheng and Chiew's experiment, the 286 uniform particle size d ¼ 1.02 mm was used, and the velocity V s 287 (injection) was measured.Kavcar and Wright (2009) conducted 288 similar experiments with both injection and suction using sediment 289 particles of d 50 ¼ 0.5 mm.Liu and Chiew (2012) observed the criti-290 cal shear stress for sediment with a median diameter of 0.9 mm 291 in the presence of downward seepage.Fig. 2 shows that the agree-292 ment between the measured and predicted critical shear stress is 293 acceptable.
302 out empirical treatment.If m ¼ 2, Eq.(16) states that upward and 303 downward V s have the same effect on τ 0 c , thereby differing from the 304 experimental data.Francalanci et al. (2008) also developed an empirical equation to express the critical shear stress under the influence of vertical velocity:

1 Fig. 2 .
shows the critical shear stress observed by Afzalimhr et al. (2007), Sarker and Hossain (2006), Shvidchenko and Pender (2000), Gaucher et al. (2010), Emadzadeh et al. (2010), Everts (1973), Graf and Suszka (1987), White (1970), Neill (1967), and Carling (1983).The observed critical shear stress largely deviates from the Shields diagram, represented by the solid line (Y ¼ V s =ω ¼ 0); the lines in Fig. 3 are results calculated using different values of Y in Eq. (14).A very small value of Y can significantly alter the critical shear stress.For example, the median sediment size in the downstream section of the Mississippi River is about 0.37 mm, and its settling velocity is about 4.52 cm=s.If the vertical velocity is 0.4 cm=s or 0.3% of the streamwise velocity, the observed critical shear stress will be 20% higher/lower than the Shields curve's prediction.Lamb et al. (2008) observed that the critical Shields stress increases 24 with a channel's slope, which is contrary to standard F2:Comparison of experimental results on threshold condition F2:2 under injection with Eq. (14).because the mobility of particles increases with a channel's slope due to the added gravitational force in the downstream direction, and vice versa.Fig. 3 explains this paradox: if a channel slope is very gentle, then a decelerating flow is likely to occur, and the upward velocity promotes sediment mobility; if a channel slope is very steep, accelerating flows are most likely, and the downward velocity constrains particle mobility. 371

383AF3: 1 Fig. 3 .
research group at Nanyang Technological University in 384 Singapore carried out a series of experiments to measure the influ-385 ence of vertical velocity on sediment transport (Cao et al. 2016).386 The tests in this study were conducted in a rectangular Perspex 387 flume that was 4.8 m long, 0.25 m wide and 0.25 m deep, supported 388 on a steel frame.Fig. 4 presents a schematic drawing of the flume.389 The sand bed, which was 1 m long and 0.25 m wide, was placed 390 approximately 3 m downstream of the upstream end of the flume.391 The test section with injection was located in the middle of the 392 sand bed, with a length ¼ 0.1 m and width ¼ 0.25 m.Holes were * Re Influence of wall-normal velocity on critical shear stress.Symbols represent measured results, and lines represent calculated results from F3:2

Fig. 4 )
Fig. 4) and was used to directly measure the bed load transport rate by collecting the sand within a known time period.The trap was connected to a PVC tube with a valve attached to it.The valve was closed during the experiment.After each test, the valve was opened and the collected sand and water drained to a container.The sand was then dried and weighed to directly calculate the bed load transport rate.Water was stored in a laboratory reservoir.Two different pumps were used to circulate flow in the flume and the injection, subsequently referred to as the main and injection pumps.The flow rates in the flume and injection areas were controlled with two separate valves and monitored using different flow meters.In order to avoid turbulence and ensure uniform flow distribution, small pipes with 20-mm diameter were installed at the entrance to the flume.

2 .
Open the sand trap valve.Turn on the main pump to a very slow flow rate.Wait until no particles are moving toward the sand trap, then close the valve and let the water slowly fill up the entire flume.The very small amount of sand collected in the sand trap at this time was not included in the determination of the sediment transport rate.3.Turn off the main pump and turn on the injection pump.Open the valve to the desired injection flow rate.4. Open the main pump and set the flow rate to 6.775 L=s; this flow rate was used for all tests.When both pumps were opened and set at their respective flow rates, the actual test commenced.5. Run the test for 30 min.During the test, keep monitoring the flow rates of both pumps to ensure that they remain constant.Videotape the bed load transport behavior in the flume.At the end of the 30-minute duration, turn off both pumps.

436 6 .Y ¼ − 1 ,F4: 1 Fig. 4 . 1 Fig. 5 .
Collect the wet sand and put it into an oven at 120°C for drying.437 Weigh the dry sand to calculate the volumetric sediment trans-438 port rate.439 The measured data (Cao et al. 2016) are presented in Fig. 5, 440 which shows that sediment transport rate can be significantly in-441 creased by an upward velocity.If the upward velocity is 80% of 442 the settling velocity (Y ¼ 0.8), then the predicted sediment trans-443 port rate can be increased 50 times g t ð0Þ.This explains why scour 444 holes are formed by an upward velocity.Fig. 5 also shows that sedi-445 ment transport rate is slightly reduced if a downward flow exists.446 If the downward velocity is equal to the settling velocity, that is, 447 the sediment transport rate will be reduced to 1=3 of g t ð0Þ, 448 because the particles are "heavier" in this case.449 Therefore, it can be inferred from Eq. (28) and Fig. 5 that up-450 ward flow dominates the sediment erosion process.In other words, 451 local scour is always associated with upward or decelerating flows 452 if the streamwise parameters are relatively unchanged.This is con-453 sistent with experimental results by Oldenziel and Brink (1974), 454 Richardson et al. (1985), and Francalanci (2006).Their experimen-455 tal results show that injection promotes sediment transport, while 456 suction reduces the rate of sand transport.Schematic drawing of flume for experiments done at Nanyang Technological University, Singapore.(Reprinted by permission from Springer: F4:2 Springer, Acta Geophysica, "Injection effects on sediment transport in closed-conduit flows," D. Cao, Y.-M.Chiew, S.-Q.Yang, © 2016.)F5:Comparison of predicted and measured sediment discharge ver-F5:2 sus vertical motion; Y ¼ 0 = no vertical velocity; Y < 0 = accelerating F5:3 flow in which sediment discharge is reduced; and Y > 0 = decelerating F5:4 flow in which sediment discharge is increased significantly.

27 is 28 .
responsible for this phenomenon.Similarly, specially designed artificial floods with high Y can flush more sediment to the sea, it is more important for the lower reach of Yellow River in order to save floodwater in the arid region It should be highlighted that an upward velocity or positive Y can in some cases trigger large vortexes, which can in return lift large and heavy particles to the water surface.Similar phenomena can be observed in powerful typhoons, hurricanes, and cyclones.All of these phenomena are initially formed by warm 29 and upwardmoving fluid (moist air or water) in which pressure becomes lower.Fluid from surrounding areas of higher pressure pushes 30 into the low-pressure area; the surrounding fluid swirls in to take the place of the low-pressure fluid.The whole system of fluid starts to spin and grows.Once formed, a system of spinning fluids can easily tear a river bottom 31 , houses, or ships into pieces; the sizes and fluids of the vortexes are largely different, but their effects and mechanism are quite similar; that is, positive Y is responsible for all these phenomena.

33 ,
in upflows or decelerating flows.Similarly, more particles are deposited on the bed in downflows or accelerating flows; thus, one can infer that the occurrence of maximum sediment concentration always lags behind the appearance of peak flow.This inference could be extended to estuaries and coastal waters in which the highest sediment concentration always occurs when streamwise velocity u ≈ 0, because the highest velocity (shear stress) corresponds with lower sediment concentration.All these phenomena indicate that sediment transport cannot be fully expressed by streamwise parameters alone, and vertical motion should be included.Discussion This paper considers why the extension of existing sediment transport formulas to nonuniform and unsteady flows is invalid.In addition, the investigation shows that vertical motion is responsible for the invalidity.In the classical theory of sediment transport, the lift force is included as the vertical motion, but it is induced by streamwise motion, and the vertical force is always upward.It becomes zero when the streamwise velocity ¼ 0. By contrast, this study highlights that vertical motion can be upward or downward and can be independent of the streamwise motion.Hence, sediment transport can be modeled in a more realistic way, especially when sediment particles are transported by waves or are subject to groundwater seepage.After the introduction of vertical velocity into existing sediment transport theory, many odd phenomena become understandable, as discussed previously.For example, the formation of ripples/dunes can be attributed to instantaneous coherent structures.Its upward velocity or ejection causes severe local scour, but scour holes with the same scour depth cannot be formed by downward velocity during its sweeping period in the bursting phenomenon 34 .In other words, the effect of sediment transport by upward or downward velocity as shown in Fig. 5 is responsible for the bedform formation.It is reasonable to assume that for large particles Y ≈ 0, vertical velocity 35

Fig. 6 .
(a) Smooth beach without scour holes generated by waves dur-F6:2 ing low tide on a beach in Wollongong, Australia; and (b) receding F6:3 waves for which water velocity is zero at the highest point, then accel-F6:4 erating down until the point where another wave is met 36 . .S., S. Wartel, B. V. Eck, and D. V. Maldegem.2005.check of online databases revealed a possible error in this reference.The issue has been changed from '8' to '9'.Please confirm this is correct.

Table 1 .
Relationship between vertical velocity and sediment transport