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This paper makes an attempt to answer why the observed critical Shields stress for incipient sediment motion deviates from the Shields curve. The measured dataset collected from literature show that the critical Shields stress widely deviates from the Shields diagram’s prediction. This paper has re-examined the possible mechanisms responsible for the validity of Shields’ diagram and found that, among many factors, the vertical velocity in the sediment layer plays a leading role for the invalidity of Shield’s prediction. A closer look of the positive/negative deviation reveals that they correspond to the up/downward vertical velocity, and the Shields diagram is valid only when flow is uniform. Therefore, this diagram needs to be modified to account for hydraulic environments when near bed vertical velocities are significant. A new theory for critical shear stress has been developed and a unified critical Shields stress for sediment transport has been established, which is valid to predict the critical shear stress of sediment in both uniform and nonuniform flows.

The initiation of sediment motion is one of the most important topics in sedimentology and geomorphology as well as hydraulic/hydrological engineering. Generally, there are two methods available in the literature to express the incipient motion that is the shear stress approach and velocity approach [

where

where ν is the kinematic viscosity of the fluid.

The original Shields diagram has been reproduced and modified by many researchers. A comprehensive review has been done by [

Some of them believe that the definition of the incipient motion may cause the invalidity of Shields diagram as the experimenter is very difficult to define precisely the status of sediment particles. Consequently, the incipient motion depends more or less on the experimental observers’ subjective judgment. To address this, criteria like “individual initial motion”, “several grains moving” and “weak movement” have been introduced to express the incipient motion [

Other researchers attribute the large discrepancy to the stochastic nature of turbulence and sediment itself, in which grain shape, orientation, exposure, protrusion, etc. can affect the critical Shields stress for example [

Over eight decades, the incipient motion of sediment transport has been extensively investigated again and again [

Probably, [

A group of researchers make some attempts to explain the large discrepancy between predicted and measured critical shear stress using channel’s characteristics, such as the channel shape and bed slope [

Therefore, the brief literature review shows that more research is needed to clarify why the critical shear stress for sediment motion depends on channel-bed slope and nonuniformity, and why Shields diagram cannot predict the critical shear stress well. The primary objectives of the present study are to 1) investigate why sometimes the Shields curve cannot express the incipient motion of sediment transport; 2) explain the dependence of Shields number on the variation of water depth along the open channel flow or channel slope; 3) establish a universal relationship between τ_{*} and the particle Reynolds number; and 4) verify the newly established equations using data from the literature.

In this study, we hypothesized that the upward/downward velocity in sediment layer caused by nonuniformity or seepage plays an important role for the invalidity of Shields’ diagram. It is ubiquitous as shown in

where ω = particle’s settling velocity in still water and ω' = the net falling velocity subject to the vertical velocity, V_{b} in the sediment layer. A spherical particle’s falling velocity ω in still water (V_{b} = 0) can be expressed as:

where d is the particle diameter, C_{d} is the drag coefficient which is dependent on the Reynolds number that is_{b} is the same as the particles’ falling velocity ω, the net vertical velocity of the particle becomes zero, thus the particle can be suspended in the flowing water like a neutrally buoyant particle. In such case, it is impossible to expect that the Shields diagram is valid to predict the particle’s critical shear stress. Likewise, if the particles in

The above simple discussion clearly shows that the presence of vertical velocity V_{b} in a sediment layer could

lead to the invalidity of Shields diagram. This inference has been confirmed experimentally by many researchers like [

In this study, the reduction of settling velocity as shown in Equation (3) can be achieved by introducing “apparent sediment density”. As the presence of vertical velocity can alter the net settling velocity, the upward velocity promote sediment’s mobility, thus the critical shear stress is reduced with seepage similar to the critical shear stress of lightweight material without seepage, even other parameters like particle sizes are identical. Similarly, the downward velocity increases the net sediment falling velocity or its effect can be alternatively expressed by increasing the apparent density, subsequently its stability.

The introduction of apparent sediment density is similar to the treatment of [

where

Equation (6) shows that if V_{b} is equal to zero, then _{b} is upwards or positive then _{b} = ω, then _{b} is downward or negative, the apparent density of sediment is higher than the density of natural sands, or the sediment’s stability is increased like heavy metals.

As the vertical velocity V_{b} in

Substituting Equation (6) into Equation (7), one obtains:

or

where

Equation (8) or Equation (9) generally expresses the influence of vertical velocity V_{b} on the critical shear stress, and it is useful to show why the vertical velocity, V is ubiquitous in open channel flows and how the vertical velocity can be induced by nonuniform flows.

In natural streams and laboratory flumes, the uniform flow is very rare, most of them are nonuniform. As shown in

where

Equation (11) shows that accelerating flows that is

where Q = discharge; U is the depth-averaged streamwise velocity, h is the water depth and b is the channel width. In Equation (12), Q/b could be constant if both Q and b remains unchanged in x direction, thus one has:

The vertical velocity

where

Equation (15) shows that the decelerating flow generates

[

Similarly, d_{*} needs modification by introducing the apparent density with the following form:

Inserting Equation (6) into Equation (17), one has

or

Therefore, the empirical equation of Shields curve by [

For the fall velocity, many empirical equations are available in the literature. [_{d} in Equation (4) with the particle diameter d_{*} and obtained the following explicit form:

It should be clarified that in Equation (3), only the real settling velocity is needed, so the d_{*} in Equation (21) should not be replaced by

Although the incipient motion driven by a uniform flow has been extensively investigated, to the authors’ knowledge no one investigates the influence of vertical velocity on incipient motion of sediment particles, this vertical flow may not be measurable or large enough to induce discernible seepage, thus it is useful to estimate the vertical velocity using some measured parameters. The average vertical velocity can be written in a similar way as shown in Equation (15) that is:

where U, the depth-average streamwise velocity, is a measurable parameter, rather than

The near bed vertical velocity that influences sediment incipient motion can be induced by either the groundwater or the surface variation as shown in Equation (22), the joint effect can be obtained by assuming it is proportional to V and the nominal seepage velocity V_{s} that is

where λ and λ_{s} are the coefficients, V_{s} = nominal seepage velocity defined by Darcy (V_{s} = ki, k = hydraulic conductivity, i = hydraulic gradient), ε_{0} = porosity of granular materials. All parameters in Equation (23) are measurable.

Generally in laboratory flumes the second term of Equation (23) is negligible that is V_{s} = 0, but in natural streams both the river flow and underground water flow can generate the velocity at the river bed, thus two terms co-exist in Equation (23).

To verify whether Equation (20) is applicable to nonuniform flows, we have comprehensively compiled 329 datasets from [

The experiments by [_{50} from 0.127 to 1.79 (mm) and specific gravity of 2.65, and 11 runs with d_{50} from 0.09 to 0.18 (mm) and specific gravity of 4.7. _{50} = 0.91 to 4.36 (mm).

[_{50} = 0.63, 1.02 and 1.95 (mm) were used, and the seepage velocity (injection) was measured with a range between (0 - 0.0138) (m/s). They found that the upward seepage reduces significantly the critical shear stress required by Shields curve. [_{50} = 0.16, 0.5 and 1.2 (mm)

Researchers | d_{50} (mm) | S | h (m) | U (m/s) | u_{*} (m) | ω (m/s) | No. of data sets | |
---|---|---|---|---|---|---|---|---|

[ | 5 - 29.1 | 0.01 | 0.03 - 0.192 | 0.28 - 0.35 | 0.029 - 0.165 | 0.685 - 1.18 | 0.13 - 0.62 | 59 |

[ | 0.016 - 2.2 | 0.02 | 0.02 - 0.07 | 0.0018 - 0.232 | 0.0062 - 0.045 | ----- | 0.00023 - 0.174 | 26 |

[ | 0.09 - 1.79 | 0.005 | 0.0094 - 0.09 | 0.1312 - 0.38 | 0.018 - 0.043 | 0.39 - 1.79 | 0.007 - 0.156 | 35 |

[ | 62 | 0 | 0.213 | 0.163 | 0.141 | ----- | 0.94 | 2 |

77 | 0 | 0.226 | 0.124 | 0.315 | 1.05 | |||

[ | 12.2, 23.5 | 0.0075, 0.025 | 0.102 - 0.2 | 0.23 - 1.6 | 0.087 - 0.155 | ----- | 0.41 - 0.58 | 9 |

[ | 0.63 - 1.95 | 0.01 | 0.027 - 0.076 | 0.09 - 0.399 | 0.017 - 0.032 | 0.02 - 1.048 | 0.08 - 0.163 | 50 |

[ | 1.5 - 12 | 0.0019 - 0.0287 | 0.002 - 0.65 | 0.1 - 1.07 | 0.026 - 0.1157 | ----- | 0.14 - 0.41 | 21 |

[ | 0.64 - 1.02 | 0.00026 - 0.00063 | 0.089 - 0.214 | 0.3 - 0.59 | 0.019 - 0.024 | ----- | 0.084 - 0.1135 | 19 |

[ | 8 | 0.0075, 0.015 | 0.13 - 0.21 | 0.726 - 0.86 | 0.05 - 0.061 | 0.362 - 0.535 | 0.338 | 9 |

[ | 0.16, 0.5, 1.2 | 0.01 | 0.23 - 0.29 | 0.23 - 0.412 | 0.013 - 0.022 | 0.67 - 1.84 | 0.019 - 0.124 | 16 |

[ | 0.8, 1.3, 1.8 | ±0.7, ±0.9, ±1.25, ±1.5 | 0.146 - 0.25 | 0.15 - 0.44 | 0.007 - 0.021 | 0.078 - 2.9 | 0.097 - 0.156 | 72 |

[ | 0.91 - 4.36 | 0.01 | 0.125 - 0.14 | 0.29 - 0.56 | 0.021 - 0.038 | 0.405 - 0.907 | 0.105 - 0.245 | 6 |

[ | 0.9 | 0.01 | 0.12 - 0.14 | 0.28 - 0.35 | 0.0215 | 0.98 - 1.71 | 0.105 | 5 |

and the observed value of seepage velocity that is V_{s} is range between (−0.0026 - 0.00223) (m/s). [

Nineteen flume experiments from [_{50} = 8 (mm) was used for their observation. Different from the prediction of [_{50} = 0.8 and 1.3, 1.8 (mm) was used for a total 72 data sets, in order to achieve nonuniform flow conditions, negative and positive bed slope (±0.7%, ±0.9%, ±1.25% and ±1.5%) were used. They found that the critical shear stress and Shields parameter for incipient motion in accelerating flow are higher than those predicted by Shields in uniform flow while their values in decelerating flow are considerably lower than that in accelerating flow.

These data are plotted in the form of the Shields diagram based on their original definitions as shown in

In the literature, many researchers attribute the deviations from the Shields curve to channel’s slope. For example [

where ϕ = angle of stream wise bed slope, θ = angle of repose. Equation (24) shows that the Shields number decreases with the increase of channel slope.

However, the formula given by [

where X = 0.407ln(142S), and the slope S is in the regime 10^{−}^{4} < S < 0.5.

_{*} could be largely different even the sediment and channel slope are set to constant. Hence, one can conclude that the invalidity of Shields prediction cannot be simply explained by the dependence of channel slope, and more research works need to be carried out for the discrepancy.

The modified Shields number in Equation (7) should remain unchanged if the apparent sediment density is introduced that is:

Using Equation (6), one obtained the critical shear stress in the following form:

_{s} is found to be 8.5. The good agreement between the measured and predicted critical shear stress indicates that the introduction of apparent sediment density is acceptable.

Different from Equation (27), [

where V_{sc} is the critical seepage velocity in a quick state and m = 1 ~ 2 and depends on the characteristics of sediments, they proposed the following equation to express V_{sc}:

It is clearly seen that Equations (27) and (28) are functionally similar to each other, and both Equations (27) and (28) give _{s} = 0.

[

Comparing Equations (27), (28) and (30), one can find that the conditions for _{s} = ω; V_{s} = V_{sc} and

It is interesting to compare these three pre-requites for _{s}) is equal to the settling velocity that is_{sc} calculated from Equations (29) and (31) is greater than ω, it implies that streamwise force is still needed to initiate the particles’ movement even if all particles are in a suspended state, which is totally unacceptable; 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, which is also impossible. Therefore, only Equation (27) gives a reasonable condition for_{sc} and Equation (31) can be obtained from Equation (29) by assuming ε_{0} = 0.

To ascertain whether the deviations from Shields curve are caused by the nonuniformity, the data without seepage in

where _{f} are the bed and energy slopes, respectively. Manning coefficient (n) can be assessed using the Strickler’s formula:

The energy slope S_{f} in Equation (32) can be determined from the Manning equation using the hydraulic radius R that is:

For all data without seepage listed in

the accelerating flow increases particles’ stability, and decelerating flow increases sediment’s mobility. In

To investigate whether all data points shown in _{b} is caused by the nonuniformity in the main flow. Therefore, Equations (22) and (23) can be simplified as follows:

Experiments by [

In these studies, flow discharge Q or mean flow velocity U; flow depth h or hydraulic radius R; median sediment size d_{50}; bed slope S_{0} were comprehensively measured. The experimental data sets from the collected data related to the nonuniform flow are plotted in _{50}.

_{s}/ω = 0), all data points can be covered by Equation (8) or Equation (9) when the parameter Y is introduced.

As mentioned, some researchers found the dependence of the critical shear stress on the channel slope [

This study re-examined the data available in the literature and found that the deviation from the shields curve could be caused by the vertical velocity, the Shields curve is valid only when the flow is uniform, which implies that the vertical velocity is zero. As the true uniform is very rare in laboratory or nature, thus it is understandable why Shields curve is invalid to express most of observed critical shear stress. Hence, a natural question to ask is whether the dependence of τ_{*} on the channel slope is also caused by the vertical flow or flow’s nonuniformity.

Equations (22) and (23) indicate that in almost all cases, there always exists the vertical velocity caused by either the exchange of groundwater and river water or the flow’s nonuniformity. Therefore, the widely observed dependence like those reported by [

We took [

This paper investigates why the observed critical shear stress widely deviates from the Shields curve. Its validity

Researchers | d_{50} (mm) | _{ } | _{ } | Flume length (m) |
---|---|---|---|---|

[ | 6.2, 8.5, 10.6, 20, 23.8, 29.1, 5, 16, 6.4 | 0.04 - 0.06 | 184.3 - 4800 | 5 |

[ | 7.95, 2.5 | 0.05, 0.05 | 638, 112 | 15 |

[ | 3.57, 1.79, 0.895, 0.508, 0.395, 0.254, 0.127, 0.18, 0.09 | 0.018 - 0.07 | 1.3 - 162 | 16.8 |

[ | 22.5, 12, 6.4 | 0.0386 - 0.1178 | 20 | |

[ | 0.9, 1.5, 1.8, 3.3 | 0.021 - 0.047 | 12 - 127 | 8 |

[ | 0.42 | 0.02 | 8.7 | 4 |

1.3 | 0.047 | 72 | ||

[ | 12.2, 23.5 | 0.05 - 0.07 | 800 - 5000 | 16.8 |

[ | 1.83 | 0.03 | 61 | 23 |

1.83 | 0.036 | 12 | ||

0.67 | 0.023 | 332 | ||

5.28 | 0.037 | 115 | ||

[ | 5.3 | 0.02 | 219 | 7.9 |

[ | 1.5, 2.4, 3.4, 4.5, 5.65, 7.15, 9, 12 | 0.025 - 0.065 | 40 - 2000 | 6.5 |

or discrepancy could be caused by many factors like sediment gradation, shapes, channel-bed slopes, measurement errors, turbulence, etc. However, this study found that the vertical motion plays an important role to initiate sediment motion or bed load transport; the vertical velocity could be induced by seepage, nonuniformity and turbulence alike. By re-examining 329 datasets available in the literature, we can draw the following conclusions:

1) The upward velocity promotes sediment mobility and downward velocity promotes sediment stability. The sediment’s mobility or stability can be equivalently expressed by its apparent sediment density which is able to eliminate the effect of vertical velocity as shown in Equation (6). This provides a useful tool to simplify the influence of nonuniformity on the incipient motion of sediment.

2) There exists vertical velocity on the channel bed and this vertical velocity could be induced by seepage or nonuniformity, the amount of the vertical velocity may be very small, but its influence to sediment incipient should not be underestimated. The joint effect can be expressed by Equation (23). For nonuniform flow, Equation (6) indicates that the sediment tends to move in decelerating flows, but it becomes more difficult to move in accelerating flows.

3) The Shields diagram is valid only when the flow is uniform, but after the introduction of apparent sediment density, the Shields diagram could be extended to express complex flows. An improved critical Shields stress for sediment transport has been established, that can predict the critical shear stress in both uniform and nonuniform flows well.

4) A new parameter Y has been proposed to express the influence of nonuniformity or seepage, this parameter should be included in the models of sediment transport. According to available experimental data in the incipient motion in nonuniform flows or in the seepage cases, good agreements between the measured and predicted values can be achieved if Y is included in the existing model, but more research is needed to determine the coefficients λ_{s} and λ in Equation (23), they could be a function of sediment gradation and shapes, turbulence, etc.