(1)
J. Aberle1, A. Dittrich2
Dr.-Ing., Dr.-Ing. habil.
1National
Institute of Water & Atmospheric Research (NIWA)
PO Box 8602, Christchurch, New Zealand
Tel.: +64-3-343-7801, Facsimile :+64-3-348-5548, E-mail:j.aberle@niwa.cri.nz
2Institute
of Water Resources Management, Hydraulic and Rural Engineering,
University of Karlsruhe
Abstract:
In this study, the recently introduced roughness
parameter s (standard deviation of the roughness heights, see Aberle et al.,
1999) is used for the determination of incipient motion in steep streams.
Relationships for the critical conditions following the Shields-parameter and
the Schocklitsch type of approach are modified with the use of experimental
data. The formulated relationships including the geometrical roughness
parameters are independent of any grain-size. This is in agreement with the
results of Parker et al. (1982), who found that the beginning of motion of bed
material with a wide grain-size distribution is not dependent on only one grain
size.
Keywords: incipient motion, roughness parameter, steep streams, river morphology
The problem of defining critical flow conditions associated with the problem of incipient motion of sediments is of fundamental importance concerning stable channel design and protection against erosion and scour. Until today especially for steep streams, no universal formula for the determination of the critical conditions has been found. The reasons for this can be seen in the influence of the morphology of steep streams, being characterised by steep slopes, large roughness elements and bed-forms (e.g. step-pool structures).
In general, the relationships for determination of critical flow conditions include a characteristic grain size. However, several studies (e.g. Parker et al., 1982) suggest, that the critical discharge for incipient motion of bed material which is characterised by a wide sediment range, is independent of the grain-size distribution, i.e. all grain sizes are entrained approximately at the same time. Patel & Radju (1999) recently confirmed this result. Following the observations of Rosport (1997), this assumption holds as well for steep streams, in which the erosion of small grains leads to movement of larger grains, which are moving into a more stable position.
Taking this into account, it was concluded that the roughness structure plays an important role on the critical conditions (Dittrich et al., 1994). Aberle et al. (1999) showed the significance of the roughness structure for the determination of flow resistance in steep streams by defining a new roughness parameter, the standard deviation s of the roughness heights. This roughness parameter is calculated by analysing longitudinal profiles and is more accurate than a characteristic grain-size, because it contains direct information about the bed geometry. In this paper, the influence of the standard deviation of the roughness heights on the critical conditions will be illustrated and discussed.
Two different types of approaches can be found in the literature to determine torrent bed stability. On one hand, the approaches are based on the determination of a critical water depth (Shield’s approach) and on the other hand on the determination of a critical specific discharge (Schocklitsch approach). Both types will be reviewed briefly in the following with respect to steep streams.
Following Graf (1991) and Bathurst et al. (1982), the classical Shields-parameter should be modified for steep streams by introducing the term [cosa tanj-sina], which represents the influence of the slope on the Shields-parameter:
(1)
with t*',cr =modified critical Shields-parameter, rs=sediment density, r=water density, g=acceleration due to gravity, d=grain diameter, a=angle of the slope, j=angle of repose and t0c=critical shear stress.
Fig. 1 Modified Shields-parameter t*',cr as a function of the relative submergence h/d50 and the slope S (Bathurst et al. 1982); the number following the symbol denotes the d50
Using data from several authors and plotting t*',cr as a function of the relative submergence h/d50, Bathurst et al. (1982) developed a stability diagram which is shown in Fig.1. In Fig. 1, for experiments with the same bed material (solid lines), t*',cr decreases with increasing submergence h/d50. However, the data does not collapse to a single line, indicating the influence of further parameters, e.g. bed geometry, angle of repose, material density, lift-force, etc.. The lift-force especially depends on turbulence characteristic and, therefore, is difficult to determine and its influence has been mostly neglected. Further statements concerning the influence of the lift-force on the Shields-parameter can be found in Dittrich (1998). The dashed lines in Fig.1 indicate data-points, which are characterised by the same slope. The theoretical curve shape of these lines, derived from Eq.(1) is shown in Fig.2 for different slopes S. The dashed lines, however, show deviations from the theoretical shape and the lines are not free from discontinuities, indicating the existence of further parameters of influence, as concluded above.
Furthermore, Bathurst et al. (1982) found that with decreasing channel slope there is a flatter relationship between the Shields-parameter and relative submergence (see Fig. 2). For slopes less than S=1%, Bathurst et al. identified a value for the Shields parameter between 0.04 and 0.06, which is the value for the classical Shields-parameter. However, following Fig.1 and Fig.2, it is difficult to define threshold values for t*’,cr for steeper slopes. Thus, the general applicability of the Shields-parameter can be questioned for mountain streams with steep slopes.
Fig. 2 Theoretical relationship between t*',cr and h/d for different slopes following Eq.1 (rs=2650 kg/m3)
Schocklitsch (1962) derived a further type of approach for the determination of incipient motion. This type of approach follows from dimensional analysis, and, different to the Shields-approach, estimates the critical specific discharge as a function of the slope and the density. The original approach given by Schocklitsch is:
(2)
Following Bathurst et al. (1982), the Schocklitsch approach is more suitable for the determination of the critical conditions in mountain streams with slopes S>2% than the Shields-approach. Bathurst et al. derived the following relationship:
(3)
There are more approaches quoted in the literature, which show the same form as the approaches following Eqs.2 and 3 do. Most of these approaches follow the relationship:
(4)
Combining the Manning-Equation, the equation of continuity and using the definition of the Shields-parameter the exponent of the slope of –7/6 can be derived directly (see Whittaker & Jaeggi, 1986).
For data obtained during the investigations of Rosport (1997), Koll & Dittrich (1998) and described in detail by Aberle (2000), the results of Fig.3 are obtained if the dm is used as a characteristic grain diameter. In the experiments, two different sediment mixtures with different maximum grain-diameters dmax were used (n: dmax=32mm and o: dmax=64 mm). The points marked with X (dmax=64mm) denote experiments, in which a slightly different procedure was applied to obtain the results. In Fig.3, a dependency on the used bed material can be identified, leading to different relationships for the two used bed materials.
Fig. 3 Resulting plot for the schocklitsch approach with data from karlsruhe university, qcrit=discharge at which the respective beds armoured
However, combining the whole data yields the following relationship:
(5)
with a mean absolute deviation of 0.26.
The above statements lead to the conclusion that further parameters, which should include information about the surface texture, are necessary to improve both types of approaches. A parameter for this purpose can be identified as the standard deviation s of the roughness heights. Aberle et al. (1999) and Aberle (2000) already showed the applicability of s as a roughness parameter for the determination of flow resistance. In this paper, it is briefly shown how this parameter can be included into the two types of approaches.
Following Aberle (2000), the introduction of the additional roughness term standard deviation s leads to a further modification of the Shields-parameter:
(6)
This modification results from the assumption that grains are embedded into the substratum and that the area A of a grain, which is exposed to the flow is not proportional to d2 but proportional to sd. The standard deviation is therefore used as a vertical length, the grain diameter as the horizontal length. Finally Eq.(5) results from the force equilibrium by neglecting the influence of the lift-force. Analysing t*m,c as a function of the slope and the parameter s/d yields the following relationship, whereas as characteristic grain size the mean diameter dm is chosen:
(7)
Eq.7 shows, that the parameter dm can be cancelled. Therefore, the relationship following Eq.7 can be reduced to hcrit/s=fct.(sina) and the following relationship, which is shown by Fig.4a, is obtained:
(8)
It has to be mentioned, that for the data used, the mean water depth was calculated as q/um, q denoting the specific discharge and um denoting the mean velocity. Interestingly, the relationship of Eq.7 is independent of any grain diameter. However, as mentioned above, following the results of Parker et al. (1982) the beginning of motion can be seen as independent of a grain size. Thus, Eq.(7) is in agreement with this finding. The mean average deviation for the relationship in Fig. 4a is 0.07, showing the goodness of the fit.
Fig.
4 (a) The ratio hcrit/s as a function of sina and
(b) Plot of qcrit/(gs3)0.5 as a function
of sina
A similar result is obtained for the Schocklitsch type of approach. As before, the grain diameter is no longer found to be relevant if an analysis following the dimensional relationship qcrit/(gs3)0.5=fct(sina, s/d) is performed:
(9)
This relationship and the accompanying data are shown in Fig.4b. In comparison to Fig.3, the dependency on the type of bed material disappears if s is used as a roughness parameter. The reduced mean absolute deviation of 0.11, in comparison to 0.26 from Fig.3, shows the applicability of s as a parameter for the determination of beginning of motion in steep streams.
In this paper, the applicability of a
statistical derived roughness parameter is investigated in order to be suitable
for the determination of beginning of motion in mountain streams. By analysing
experimental data, it was found that the standard deviation s of the roughness
heights leads to improved results for the determination of incipient motion. The
scatter of experimental data can be reduced remarkably by introducing s into
relationships following the Shields-parameter or following a Schocklitsch type
of approach. The findings confirm, that the beginning of motion of a wide
grain-size distribution is not dependent on only one grain-size. Further
research is needed to find a relationship between the grain-size distribution
and the geometry of the river bed. This relationship should exist, because the
bed-surface consists of gravel of a certain mixture. However, the investigations
of Aberle (2000) and Nikora et al. (1999) showed, that this relationship cannot
be derived easily.
Acknowledgements
This study was conducted under the support of the DFG-Graduiertenkolleg “Ökologische Wasserwirtschaft” (Ecological Water Resources Management) at the University of Karlsruhe and the EROSLOPE-Project (ENV4-CT96-0247). Graeme Smart provided suggestions on the manuscript.
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