RELACION BENEFICIO COSTO

Threshold of motion data have been evaluated, traditionally, using values of shear stress calculated from other flow parameters, i.e. rather than by direct measurement. It is proposed that previously excluded wave period dependency, in relationships for the wave friction factor ( fw) causes the observed pattern, simply as a result of the ‘incorrect’ calculation of

τcrit. An alternative method for the calculation of fw in the rough-transitional regime is

described and the effect on δτ0 crit /δT is discussed, below.

A broad range of empirical relationships is available to describe fw ,as a function of flow and sediment parameters over (flat) rough beds. Typically, the (analytical) laminar solution is used in relation to laminar flow conditions whereas various fully turbulent solutions are used for both transitional and turbulent conditions (e.g. Jonsson, 1966; Kamphuis, 1975; Justesen, 1988; Voulgaris et al., 1995, and references contained therein). In these diagrams, the transitional region was interpolated as a single line for given a/D; data and observations in this region are considered to demonstrate significant scatter (Kamphuis, 1975). As

highlighted in Voulgaris et al., none of the relationships suggested previously explain, or substantially reduce, the gradient δτ0 crit /δT; nor do they reproduce the period dependence,

observed in the transition to turbulence (as described in Chapter 6).

The friction factor diagrams presented by Jonsson (1966), Kamphuis (1975) and Justesen (1988) show fw increasing throughout the transitional region from a single point of departure from the laminar solution, i.e. assuming a single value of Re, for all T at the transition to turbulence. An apparently near-linear interpolation is used between the laminar and fully turbulent conditions (which appears curved on log-log axes). In addition, previous authors

have not reported variation in fw, as a result of T, in fully turbulent flows. Therefore, it was

assumed that any differences caused by T in the transitional region become asymptotic with the fully turbulent solution, as Re→Eq. 4.16 (see Figure 7.8). Other, more detailed studies of

fw in the transitional regime, such as those undertaken by Jensen et al. (1989) and Lodahl et al. (1998) using smooth beds, did not incorporate a sufficient range of conditions in order to observe the effect of T.

A simple numerical model has been constructed, to apply the new transition to turbulence relationships from the present study to the prediction of fw under transitional flow conditions. To this end, a number of assumptions were made, as outlined below.

That

1. Eq. 4.9 for the laminar friction factor can be applied to all (sinusoidal) laminar flows (Re ≤ Recrit).

2. That Eq. 6.1 and Eq. 6.5 represent the transition to turbulence and that this condition corresponds closely to initial deviation from Eq. 4.9.

3. That the relationship of Kamphuis (1975) (Eq. 4.11) can be used for the prediction of fw under rough turbulent flow conditions, for all values of T.

4. That the transitional region takes the form of a linear interpolation, between fw at the point of departure from the laminar solution and the corresponding rough turbulent value.

The second part of 2 (above) was made on the basis of the modified friction coefficient diagram of Jensen et al. (1989) (Figure 2.7); this suggests that (for smooth beds), initial deviation of fw from the laminar solution at the phase of peak τo occurs at the same condition

identified as transition over such beds (Re≈1.6×105

), in the present study. However, the patterns of transition were presented graphically only in this reference and no data were found to describe, either quantitatively or qualitatively, such patterns over rough beds or the variation of such a pattern with T. The initial departure from the laminar solution for given

a/D over a range of T was calculated then using the following model approach outlined below.

1. A value of fw was chosen and the corresponding value of a/D for rough turbulent flow was calculated using Eq. 4.11.

2. An arbitrary starting value of D (close to the expected final value) was used, to establish the corresponding value of a.

3. A value of T was selected and linear wave theory used to calculate U∞ and, subsequently, Re, using ν=10-6

.

4. Using Eq. 6.5, the parameters a/D and T or D were used to calculate a second value of Re, corresponding to the predicted transition to turbulence. At this stage, the two values of Re are not the same, but vary at different rates with the value of D (Figure 7.7).

5. Using solution-finding software, the variable D was then altered incrementally, until the two values of Re were identical, i.e. the combination of flow and grain

parameters corresponded simultaneously to the predicted transition to turbulence, also, to the selected value of a/D. The corresponding value of fw was calculated subsequently using Eq. 4.9.

6. Stages 2-5 were then repeated for other values of T.

0 200 400 600 800 0 0.5 1 1.5 2x 10 5 (a) Grainsize, D (μm) Re 0 500 1000 1500 0 1 2 3 4 5x 10 4 (b) Grainsize, D (μm) Re

Figure 7.7. Variation of Re with D in the wave friction factor model (T=6s): (a) [fw=0.01 (a/D=545.7), R=0.5]; (b) [fw=0.02 (a/D=118.8), R=0.5]. Key: blue line – flow Re (stage 3); red line –the transition to turbulence.

The resulting effect on the wave friction factor, of period dependence in the transition to turbulence, is shown in Figure 7.8; this was constructed on the basis of the diagram of Kamphuis (1975). Shown also in the Figure are: the analytical solution for laminar flows; the empirical relationship for smooth turbulent conditions; and the empirical limit for fully developed rough turbulent flow (Kamphuis, 1975). Solutions for four values of a/D are shown, corresponding to four (constant) values of fw in rough turbulent flow conditions; the selected values reflect the range of flow and sediment parameters upon which the transition to turbulence relationships were empirically-based.

102 103 104 105 106 107 10−3 10−2 10−1 100 Re f w a/D=12.0 1449.2 545.7 118.8 Increasing T [T=2,4,6,8,10,12,14s] Eq. 4.9 Eq. 4.16 Eq. 4.17

Figure 7.8. The effect of wave period on the wave friction factor coefficient. Coloured arrows indicate the effect of increasing T. See text for more details.

Over rough beds, the value of Recrit for initial transition to turbulence varies with T (Eq. 6.5);

consequently, so does the value of fw at the initial deviation from the laminar solution (Eq. 4.9). At small a/D (e.g. a/D =12), calculations showed that initial departure from the laminar solution occurred at a value of Re larger than is required for fully developed turbulence. This situation cannot exist in real flows, but may arise numerically in this case, either: (a) as a result of as unaccounted for period dependence in the relationship between a/D and fw in rough turbulent flows (Eq. 4.11); or (b) as a result of the overestimation of Recrit by Eq. 6.5

(transition to turbulence), at large grain sizes. The former explanation appears to be more likely, because data within the range 2<a/D<15 were incorporated into Eq. 6.5, this then provided equally good representation of such data, when compared to larger values of a/D.

At smaller values of a/D, the range of Recrit (and fw) calculated by using this approach, for

the same range of T, was broader. Values of fw at all T were less than the equivalent rough turbulent value. Similarly, the range of Re over which transition occurred was reduced (this effect was enhanced at larger T). At very large a/D (≈>1500): the range of Recrit was reduced

to a constant value at the smooth bed limit (Re=1.66×105

), with an associated single value of

fw; predicted values of fw in the rough turbulent regime were almost equal to the laminar case initially but, subsequently, are less (reducing progressively down to the smooth turbulent

limit); and, the range of Re over which transition occurred was larger, relative to the previous example.

New values of fw were calculated for the threshold of motion data. Observations made under either fully laminar or rough turbulent conditions were interpreted, using Eq. 4.9 or Eq. 4.11 directly, as appropriate; transitional values were calculated by fitting the observed values of Recrit, a/D and D to the model described above. The difference in the predicted fw between

the laminar and transitional solutions was calculated for each case; the results are shown in Figure 7.9. These data represent the relative increase in the calculated τ0 crit over that

calculated using the laminar solution alone. The new model typically increased the predicted value of fw at larger T and D. However, at small grain sizes (approximately D<300μm),

corresponding to large values of a/D, the friction factor in rough turbulent flow was equal to, or less than, that in the laminar case.

0 5000 10000 0 200 400 600 800 Grainsize, D (μm) % Increase in f w (b) 0 500 1000 −100 −50 0 50 100 150 Grainsize, D (μm) % Increase in f w (a) Threshold of motion under laminar or near laminar conditions

Threshold of motion under transitional conditions

Figure 7.9. Percentage difference in fw between the fully turbulent (Eq. 4.11) and laminar (Eq. 4.9) solutions at the threshold of motion: (a) D<1mm; and (b) D<1cm. Dotted arrows indicate the effect of increasing wave period.

It was anticipated originally that δτ0 crit /δT might be reduced through the use of the new

model by causing fw, hence, τ0 crit, to increase at a greater rate under longer wave periods

through the transitional regime. However, this effect was offset by an increase in a/D; this was associated also with increasing wave period, at threshold. Although δτ0 crit /δT was

reduced in some cases, the new model had generally the opposite effect, producing a mean increase in the gradient steepness of 25%; there was also a corresponding increase in the

scatter of predicted τ0 crit (see Figure 7.10), which somewhat negated the significance of any

change in the observed gradient.

2 4 6 8 10 0 0.5 1 1.5 τ 0 crit (Nm −2 ) Wave period, T (s) (a) 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 τ 0 crit (Nm −2 ) Wave period, T (s) (b)

Figure 7.10. Comparison of τ0 crit for threshold of motion under sinusoidal flows, calculated

using the laminar solution (O) and the transitional model (Δ) for (a) [D=1800μm, ρs=2600

kgm-3]; and (b) [D=800μm, ρs=2630 kgm -3

].

By manipulating the pattern of δfw/δRe in the transitional region, this model could modify

δτ0 crit /δT, potentially reducing the gradient to zero. However, this would be an empirical

adjustment for the transitional region only and significant δτ0 crit /δT would still exist for

laminar flows. This approach makes the assumption also that δτ0 crit /δTshould equal zero, in

transitional or even laminar flows; this is an assumption challenged below.