The approach of Shields (1936)

The approach of Shields (1936), together with subsequent developments of this approach, are described in Section 2.14. The construction of the Shields parameter and diagram, for unidirectional and oscillatory flows, implies that for any given uniform sediment, the threshold condition is represented by a single value of τc crit. This condition precludes the

assumption that the critical value of Uc varies then, with T. However, the present study has shown that for oscillatory data, within the range investigated, the negative gradient δτc crit /δT

is significant, becoming more pronounced into the transitional/turbulent regime for observations relating to any given sediment (Section 7.3). Consequently, the positive

gradient δUc crit /δT is correspondingly smaller than would otherwise be predicted. This effect was less (closer to the assumptions of Shields), but was still clearly present in those data that were deemed to have been collected under laminar flow conditions; it increased as the flow became progressively more (transitionally) turbulent.

The direct result of this failure in the assumptions is to introduce scatter in oscillatory flow data about the Shields curve, as was demonstrated for sinusoidal flows in Figure 2.11. Values of the Shield parameter are scattered vertically by the effect of wave period (larger values correspond to smaller T). Scatter is least when θcrit is plotted against D* (Eq. 2.42),

where the scatter is limited to differences in θcrit. However, if θcrit is plotted against Re*, then

an additional lateral scatter away from the Shields curve is included also; this is due to variation in u* (a component parameter in Re*) with T. The apparent scatter is greatest for

threshold around the transitional region of the curve, for two reasons: (1) because δτc crit /δT

and, therefore, the range of θ is larger for threshold occurring under transitional flows (in comparison to laminar flows); and (2) this larger range is further exaggerated visually by the logarithmic (vertical) scaling of the diagram.

100 101 102 103 10−2 10−1 100 D_* Shields parameter, θ Shields curve (1936) Bagnold (1946) Manohar (1955)

Hammond and Collins (1979) Tomlinson (1992)

Present study: R=0.5 Present study: R=0.6 Present study: R=0.7

Figure 7.33. The Shields curve for threshold of motion (Eq. 2.46) and threshold data for sinusoidal and asymmetric oscillatory flows.

The effect of flow asymmetry has been determined on the basis of the relationship between R

and τc crit (Section 7.4). The data collected during the present study are plotted, in relation to

the Shields curve and sinusoidal data from previous investigators, in Figure 7.33. In addition to the effect of T, previously described, the effect of asymmetry is to increase further the value of θcrit and Re* (if used). This increase is consistent over the range of flow/sediment

conditions investigated. For the same reasons discussed above, the apparent additional scatter is relatively greater when θcrit is plotted against Re*, rather than D*. In the Figure, the

region; however, it appears to remain nearly constant because the decrease is balanced (visually) by the logarithmic vertical scale.

In order to represent accurately the observations and results presented herein, the Shields curve should be plotted (preferably) using D* on the abscissa. Whereas a single curve has typically been previously used, the new results would be represented more closely by a series of similar curves, shifted from a baseline representing large T (possibly equivalent to the unidirectional case), to larger θ with decreasing T and increasing R. The magnitude of the shift should theoretically vary laterally across the plot with D*, with a smaller range under laminar flows. This range would increase gradually over the transitional regime, possibly becoming stable at some larger range within the turbulent portion of the diagram.

Alternatively, a single Shields type curve may be maintained, by changing the numerator (representing erosive forces) in the construction of θ. For laminar sinusoidal flows, the parameter τ0 crit, in θ, should be replaced with the time-mean shear stress parameter,

ω ω ω ω τ τ

τ

1.25 25 . 1 max) ( max) ( + − t t Eq. 7.7

utilising a mean characteristic time-averaging interval Δt=2.5s. This parameter will reduce the scatter relating to wave period effects within groups of data, by a factor of 2-4. Residual scatter of data groups (collected by different individuals) about the curve will still be present due to the use of a mean value of Δt; this is primarily representative of differences in the methodologies used. In the turbulent regime, the shear stress should be replaced altogether, with a parameter reflecting U∞ crit. In order to maintain θ as a dimensionless quantity, τ0 crit

may be replaced by,

c

U

_∞2

ρ

Eq. 7.8

where c is a scaling constant in order to maintain visual continuity of the curve between the transitional and turbulent portions of the diagram. Thus this parameter has the same units as

τ but is not scaled with T in the same way. In the transitional regime, the characteristic shear stress parameter lies between the laminar and turbulent solution; therefore, it should be a function of the relative presence of turbulence, at threshold (a function of the standard parameters, i.e. T, R, D and ρs).

The inclusion of flow asymmetry is more complex, as the solution is a function of both T and

R; as such, it is suggested that an altogether separate diagram be constructed. This diagram would have separate curves, that increase in magnitude of θ with decreasing T. Plotting sinusoidal or asymmetric data in the laminar/near laminar transitional regime, the parameter

τc crit in θ should be replaced with the phase-mean shear stress parameter,

° + ° − 65 65 max) ( max) ( τ τ ω ω

τ

t t Eq. 7.9

which utilises a mean characteristic phase-averaging interval, Δωt=130°. Based upon a similar approach as that used for the sinusoidal data, this new parameter will be also closely representative of the laminar regime. In the turbulent regime, Eq. 7.9 may be applied

theoretically but, due to a lack of data, it is not clear how this parameter should change under the influence of progressively greater turbulence within the transitional and turbulent

regimes.

Alternatively, as applied previously to the critical velocity approach (Section 7.7.1), the critical stroke length may be hind-cast and applied directly to the asymmetric case.

In order to incorporate the above suggestions, the curve representing the threshold of motion will have to be redrawn as a ‘line of best fit’ to the data, on the basis of the new parameters (Eq. 7.7, Eq. 7.8, Eq. 7.9 and R). Because of fundamental differences in the parameters representing laminar and turbulent flows it may be more appropriate to consider the single Shields curve approach as a numerical, rather than simply a graphical, approach to the prediction of sediment threshold.

7.7.3.The Moveability Number approach

The Moveability Number has been described in Section 2.15. The Moveability Number has been calculated for all of the sinusoidal and asymmetric data; it is plotted against D, in Figure 7.34. The implication of the ‘moveability curve’ is that, at small D, the fluid shear necessary to cause the threshold of motion is smaller, when compared to the fall velocity of the same particle within the same fluid. As such, the gradient of the curve represents flow conditions at threshold, changing from ‘dominantly laminar’ to ‘dominantly turbulent’ as the grain size increases. The location and shape of the curve (i.e. the distribution of data around the curve) is affected then by the effects of sediment density, wave period and flow

0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.2 0.4 0.6 0.8 1 1.2 Grainsize, D (μm) Moveability number, W s /u *

Quartz grains, sinusoidal flows Quartz grains, asymmetric flows

All data ρ

s<1500kgm

−3

All other data Eq. 7.9

m +/− 0.00005

Figure 7.34. The Moveability Number parameter for all sinusoidal and asymmetric threshold of motion observations used in the present study.

When plotted against grain diameter, the Moveability Number (MN) parameter followed the relationship ) / (m D

e

n

MN

=

×

Eq. 7.10

where n and m are fitting coefficients, dependant upon T, ρs and R. A general empirical

relationship was fitted to all the sinusoidal data using [n=0.1 m=4.44×10-4]. This relationship produced an error of the order ±50% when reverse calculating the corresponding value of

τccrit for all data. The majority of the error could be attributed to variation in the critical shear

stress associated with T and ρs. The lesser effect of asymmetry was also present. The

‘moveability curve’ was generally shifted away from the origin, with decreasing T or with increasing ρs and/or R. By providing a single value of τc crit, the moveability curve represents

the data no better (or worse) than any other methods, which do not include differences attributable to T or R.

Values of predicted τc crit were sensitive to any small offset, between the curve and the data

used in the fitting of the curve. Given the complex interaction of the fluid and sediment parameters in regulating turbulence and susceptibility to erosion, it does not seem reasonable to assume that all the grain sizes may be represented by a simple exponential function covering laminar, transitional and turbulent flows. However, given the reasonable estimation

of τc crit using the general solution (see above), the effect of turbulence appears to be largely

implicitly accounted for by the use of the settling velocity. For accuracy, future users of such empirical relationships will have to use standardised methods to calculate Ws, as prescribed by the original investigator; failure to do so would result potentially in a significant offset in predicted values of τc crit. Such general relationships represent the mean of the available data

and users will therefore also have to be aware of the potential offset in predicted τc crit as a

result ofρs, T and R; additional variance might be observed as a result of other stochastic

parameters such as grain pivoting angle (a result of grain shape and bed packing).

In document The effects of variation in wave period and flow asymmetry in sediment dynamics (Page 186-191)