- Email: [email protected]
Entrainment threshold of natural grains in liquids determined empirically from dimensionless settling velocities and other measures of grain size
ELSEVIER
Sedimentary Geology 119 (1998) 17–23
Entrainment threshold of natural grains in liquids determined empirically from dimensionless settling velocities and other measures of grain size J.P. Le Roux * Department of Geology, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa Received 18 March 1997; accepted 28 January 1998
Abstract Grain size plays a crucial role in sediment transport, but different methods of size analysis do not yield comparable results. Particle diameters obtained by settling tube, for example, cannot be employed in equations based on sieve sizes. Although the majority of experiments on the initiation of sediment motion has been based on sieve data, settling tube analysis is probably superior as both the settling rate and the entrainment threshold of grains are determined by the same factors of size, shape and density. In this paper, empirical equations are provided which allow the determination of a non-dimensional critical shear stress directly from the equivalent sedimentation diameter. The equations apply to the entrainment of well sorted grains with a high sphericity on flat sediment beds, and are valid for particles and liquids of different densities. 1998 Elsevier Science B.V. All rights reserved. Keywords: Reynolds number; entrainment threshold; settling velocity
1. Introduction The initiation of sediment motion is an important concept in many engineering and sedimentological studies. As a result, much work has been conducted to determine the critical shear stresses required to entrain particles of different sizes, shapes and densities in unidirectional currents (e.g. Miller et al., 1977; Miller and Komar, 1977; Yalin and Karahan, 1979; Le Roux, 1991; Bridge and Bennett, 1992). In the theoretical–empirical curves and equations derived from these studies, grain size plays a pivotal role, either being plotted directly against the critical shear Ł Fax:
C27 (1) 808-4336; E-mail: [email protected]
stress or some related parameter, or as part of a dimensionless relationship such as the grain Reynolds number Re* . However, as pointed out by Pettijohn (1975) and Winkelmolen (1982), ‘size’ is an elusive concept, because it can be expressed variously as volume, weight, surface or cross-sectional area, intercepts through particles or projections, and settling velocity. Grain sizes are commonly determined by sieve analysis and consequently expressed in terms of ‘sieve diameters’, which are construed as equivalent spheres. The actual dimensions, however, depend largely on the shape of the grains and can deviate substantially from this measure. The problem is compounded by screen imperfections and differences in sieving procedures (Komar and Cui, 1984).
0037-0738/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 7 - 0 7 3 8 ( 9 8 ) 0 0 0 2 2 - 0
18
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
Similar constraints apply to grain-size analysis by settling tube, where the dimensions are interpreted in terms of ‘equivalent sedimentation diameters’, i.e. the dimensions of spheres with the same settling rates. It is clear that the actual effects of size and shape cannot be evaluated properly without physically measuring the orthogonal axis dimensions of individual grains, but this procedure is extremely time-consuming and impractical when dealing with numerous small particles. Given a choice between sieve and settling tube analysis as a means of determining the grain size, however, one could reason that the equivalent sedimentation diameter, being a behavioural measure incorporating the properties of size, shape and density, would be preferable in entrainment studies where these factors also play a crucial role. Kench and McLean (1997) concluded that grain-size parameters obtained by sieve analysis can grossly distort the interpretation of depositional and energy processes operating within bioclastic environments, and that hydraulic-size estimates provide a much better tool. Willetts et al. (1982) also argued for adopting settling velocity in all investigations of this kind. Unfortunately, the vast majority of studies on sediment transport is based on sieve-size data, so that the latter require some transformation to conform to settling tube grain dimensions. This relationship is explored below, in order to find a simple means of predicting the entrainment threshold of natural grains directly from their measured settling velocity. 2. Previous work Attempts to relate settling velocity to sediment transport have been made in the past, notably by Collins and Rigler (1982); Komar and Clemens (1985), and Bridge and Bennett (1992). Collins and Rigler (op. cit.) conducted experiments on ilmenite, zircon, rutile and cassiterite grains, used in conjunction with comparable quartz data from White (1970). They concluded that the measured grain settling velocity .Wm / is a good indicator of the critical shear stress .−c / of a wide range of densities under hydraulically smooth boundary conditions, for which the following relationship applies: −c D 1:24 Wm0:33
(1)
Eq. 1 is valid only for water at 20ºC and Re* numbers less than 5. Komar and Clemens (1985) also noted that curves of the threshold friction velocity UcŁ versus Wm for grains of different densities converge in the range of Re* numbers studied by Collins and Rigler (1982), which explains their success in deriving Eq. 1 for minerals of contrasting densities. As Eq. 1 is of limited application, therefore, Komar and Clemens (1985) proposed a universal equation valid for a wide range of densities and liquid compositions: ½0:282 ¼ ²s ² Ł g Wm0:154 (2) Uc D 0:482 ² ² where ² s and ² are the densities of the grain and fluid, respectively, g is the gravity constant, and ¼ is the dynamic (molecular) viscosity. From UcŁ ; the critical shear stress − c can be obtained: −c D .UcŁ /2 ²
(3)
Unfortunately, Eq. 2 is also of limited use, as it is valid only for very small grains settling within the Stokes range .Re < 0:5/: Bridge and Bennett (1992) followed a more theoretical approach to model the entrainment of sediment grains, using the settling velocity .W /; the ratio between the local fluid velocity at the level of effective fluid thrust on bed load grains and the shear velocity .a/; the static friction coefficient .Þc /; the bed slope .S/ and a measure of vertical turbulence asymmetry .b/ to define the following threshold equation: s tan Þc cos S sin S (4) UcŁ D W ac2 C b2 tan Þc For hydraulically rough flows, Eq. 4 gives UcŁ ³ 0:1 W (Bridge and Bennett, 1992). The use of Eq. 4, however, requires input from several parameters which are not easily obtained, whereas the simplified relationship between UcŁ and W is highly inaccurate when compared with experimental results (Table 1). The need for a practical, universal equation or set of equations applicable to different fluids and grain densities over a wide range of Reynolds numbers is therefore evident.
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
19
Table 1 Comparison of critical shear stress determined for different quartz grain sizes in water at 20ºC, using existing formulae and equations proposed in this paper D (cm)
0.0031 0.0063 0.0125 0.0250 0.0500 0.1000 0.2000 0.4000 0.8000
Wm
Wds
0.1001 0.4017 1.3338 3.5955 7.6953 14.574 23.025 33.779 43.356
0.0339 0.1369 0.4619 1.2938 2.9577 6.2012 10.960 18.251 25.908
Wdm
0.0395 0.1585 0.5263 1.4187 3.0364 5.7507 9.0851 13.329 17.107
−c Collins and Rigler (1982)
Komar and Clemens (1985)
Bridge and Bennett (1992)
This study
0.5802 0.9177 1.3636 1.8916 2.4315 3.0020 3.4910 3.9616 4.3018
0.5505 0.8446 1.2223 1.6590 2.0971 2.5530 2.9391 3.3074 3.5717
0.0001 0.0016 0.0178 0.1290 0.5911 2.1203 5.2920 11.390 18.764
0.8492 1.2710 1.7423 2.1964 2.8541 6.6074 14.503 29.138 58.275
3. Present approach
way: r ²s ² D Dd D D 3 ²g ¼2
s
ReŁ2
In order to accommodate grains of different densities in various fluids, it is necessary to use a non-dimensional settling velocity, which is given by: s ²2 3 Wd D W (5) ¼g.²s ²/ Similarly, a dimensionless critical shear stress is required, for which the ‘Shields parameter’ is commonly employed. It is expressed as:
Using Eq. 8, the dimensionless grain size Dd for every data point on the Yalin and Karahan (1979) diagram can be determined. From these values of Dd ; the dimensionless settling velocity of sieve-size spheres can be calculated by employing the set of equations given by Le Roux (1992):
D −c =.²s
Wd D .0:2354Dd /2 forDd < 1:2538
²/g D
(6)
where D is the grain size (usually expressed in terms of the sieve diameter). Although there is a considerable data base on threshold values in the literature, only those experiments dealing with non-cohesive, well sorted grains with a relatively high sphericity entrained on flat sediment beds are considered in this study. Most of the data satisfying these requirements were collected by Miller et al. (1977) and Yalin and Karahan (1979). The latter authors included all of the data of Miller et al. (1977), with the exception of Paintal (1971) and Everts (1973). Yalin and Karahan (1979) plotted threshold data from 19 different sources in a diagram of the Shields parameter against the grain Reynolds number Re* , which is given by: ReŁ D U Ł D²=¼
(7)
Le Roux (1991) showed that Re and are related to the non-dimensional grain size Dd in the following *
Wd D .0:208Dd
3
(10)
0:37/
for 2:9074 < Dd < 22:9866 Wd D .0:8255Dd
(9)
0:0652/3=2
for 1:2538 < Dd < 2:9074 Wd D .0:2636Dd
(8)
(11)
5:4/2=3
for 22:9866 < Dd < 134:9215
(12)
Wd D .2:531Dd C 160/1=2 for 134:9215 < Dd < 1750
(13)
In Fig. 1, the data of Yalin and Karahan (1979) have been recast in a plot of against Wd using Eqs. 8–13. Also included in this diagram are the results of Paintal (1971); Everts (1973), and the data for turbulent flow of Govers (1987). For comparison,
20 J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23 Fig. 1. Plot of dimensionless critical shear stress . / against dimensionless settling velocity .Wds / of equivalent spheres based on sieve diameter. Data from Paintal (1971); Everts (1973); Yalin and Karahan (1979), and Govers (1987). The data points for hydraulically smooth conditions of Collins and Rigler (1982) are also shown for comparison.
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
the data for hydrodynamically smooth conditions of Collins and Rigler (1982) are also plotted, although they were not taken into account for the regression line. These plots correlate reasonably well with the rest of the data for turbulent conditions, but cluster below the regression line in Fig. 1. Linear regression of against Wd yields the equations: D 0:0717 log10 Wds C 0:0625 for Wds < 2:5
(14)
D 0:0171 log10 Wds C 0:0272 for 2:5 < Wds < 11
(15)
D 0:045forWds > 11
(16)
21
4. Relationship between settling velocity of sieve-diameter spheres and natural grains As Eqs. 14–16 apply to the hydrodynamic settling velocity of sieve-diameter spheres .Ws /; the relationship between this measure and the actual measured settling velocity of natural grains .Wm / in water must be established. Experimental data of Zegzhda (1934); Arkhangel’skii (1935); Sarskisyan (1958); Kennedy and Koh (1961); Mamak (1964); Schlee (1966); Sanford and Swift (1971); Baba and Komar (1981), and Raudkivi (1990), related to quarter phi sieve sizes and recast in dimensionless form, are plotted in Fig. 2. A polynomial curve fitted to these data yields a correlation coefficient of 0.9852 and shows that the dimensionless settling velocity of sieve-size spheres .Wds / is related to the dimensionless settling velocity of natural grains .Wdm / in the
Fig. 2. Plot of dimensionless measured settling velocity .Wdm / against dimensionless settling velocity of a sphere .Wds / calculated with Eqs. 9–13. Mean values for measured settling velocity from Zegzhda (1934); Arkhangel’skii (1935); Sarskisyan (1958); Kennedy and Koh (1961, table 2), Mamak (1964); Schlee (1966); Sanford and Swift (1971); Baba and Komar (1981, fig. 5), and Raudkivi (1990). In all cases grain sizes were taken as the retaining quarter phi sieve size.
22
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
following way: 2 Wds D 0:0384 Wdm C 0:8575 Wdm
(17)
Eqs. 14–16 and Eq. 17 can now be combined so that the dimensionless critical shear stress can be calculated directly from the measured dimensionless settling velocities: D
0:0321 ln Wdm C 0:0655
for Wdm < 2:6103
Acknowledgements
(18)
I wish to thank Drs. J.J. Williams, Z. Jiang and an anonymous reviewer for their critical comments, which assisted in improving this paper.
(19)
References
D 0:0087 ln Wdm C 0:0256 for 2:6103 < Wdm < 9:1108
whereas Eqs. 18 and 19 apply to settling tube data. Where grain axis dimensions are available, settling velocities can also be predicted fairly accurately using the equations of Le Roux (1996), allowing direct calculation of the entrainment thresholds.
For dimensionless settling velocities above 9.1108, is constant at 0.045. From ; the actual critical shear stress −c can be calculated using Eq. 6. The value of D is obtained by converting Wm to the dimensionless settling velocity of its corresponding sieve size (Eqs. 5 and 17), and using the inverse of Eqs. 9–13 (see Le Roux, 1992) to obtain the dimensionless grain size Dd : D is then calculated from Dd using Eq. 8. 5. Conclusions Many laboratories routinely use settling tubes to determine sediment grain sizes, which can differ significantly from that obtained by conventional sieve analysis. For example, if the equivalent sedimentation diameter (0.3654 cm) for the last record in Table 1 were substituted for the actual sieve size (0.8 cm), a critical shear stress of 26.6 instead of 58.3 g cm 1 s 2 would be obtained. Settling tube data therefore cannot be used in equations based on sieve sizes to determine entrainment thresholds. Existing equations based on settling velocity (Collins and Rigler, 1982; Komar and Clemens, 1985; Bridge and Bennett, 1992) apply only to very limited conditions and appear to be very inaccurate when compared with a wider range of experimental data (Table 1). The equations provided here are valid for grains of any density in any liquid. (For such grains in air, special conditions apply — see Le Roux, 1997.) They can also be used to determine the dimensionless critical shear stress for Re* values between 0.03 and more than 1000. For sieve data, Eqs. 14–16 can be employed (using the sieve diameter for D/;
Arkhangel’skii, B., 1935. Experimental study of hydraulic coarseness scale of particles. Izv. NIIG 15, Moscow (in Russian). Baba, J., Komar, P.D., 1981. Measurements and analysis of settling velocities of natural quartz sand grains. J. Sediment. Petrol. 51, 631–640. Bridge, J.S., Bennett, S.J., 1992. A model for the entrainment and transport of sediment grains of mixed sizes, shapes, and densities. Water Resour. Res. 28, 337–363. Collins, M.B., Rigler, J.K., 1982. The use of settling velocity in defining the initiation of motion of heavy mineral grains, under unidirectional flow. Sedimentology 29, 419–426. Everts, C.H., 1973. Particle overpassing on flat granular boundaries. Proc. Am. Soc. Civ. Eng. 99, 425–438. Govers, G., 1987. Initiation of motion in overland flow. Sedimentology 34, 1157–1164. Kench, P.S., McLean, R.F., 1997. A comparison of settling and sieve techniques for the analysis of bioclastic sediments. Sediment. Geol. 109, 111–119. Kennedy, J.F., Koh, R.C., 1961. The relation between the frequency distributions of sieve diameters and fall velocities of sediment particles. J. Geophys. Res. 66, 4233–4246. Komar, P.D., Clemens, K.E., 1985. The relationship between a grain’s settling velocity and threshold of motion under unidirectional currents. J. Sediment. Petrol. 56, 258–266. Komar, P.D., Cui, B., 1984. The analysis of grain-size measurements by sieving and settling-tube techniques. J. Sediment. Petrol. 54, 603–614. Le Roux, J.P., 1991. A rapid method to determine the critical shear stress for sphere entrainment under unidirectional fluid flow. Sediment. Geol. 75, 1–3. Le Roux, J.P., 1992. Settling velocity of spheres: a new approach. Sediment. Geol. 81, 11–16. Le Roux, J.P., 1996. Settling velocity of ellipsoidal grains as related to shape entropy. Sediment. Geol. 101, 15–20. Le Roux, J.P., 1997. Relationship between aerodynamic entrainment threshold and hydrodynamic settling velocity of particles. Sediment. Geol. 109, 199–205. Mamak, W., 1964. River regulation. Translated from Polish by U.S. Dept. of Interior and NSF. Washington, DC. Miller, M.C., Komar, P.D., 1977. The development of sediment
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23 threshold curves for unusual environments (Mars) and for inadequately studied materials (foram sands). Sedimentology 24, 709–721. Miller, M.C., McCave, I.N., Komar, P.D., 1977. Threshold of sediment motion under unidirectional currents. Sedimentology 24, 507–527. Paintal, A.S., 1971. A stochastic model for bed load transport. J. Hydraul. Res. 9, 527–553. Pettijohn, F.J., 1975. Sedimentary Rocks (3rd ed.). Harper and Row, New York, 628 pp. Raudkivi, A.J., 1990. Loose Boundary Hydraulics. Pergamon Press, New York. Sanford, R.B., Swift, D.J.P., 1971. Comparison of sieving and settling techniques for size analysis, using a Benthos Rapid Sediment Analyzer. Sedimentology 17, 257–264.
23
Sarskisyan, A.A., 1958. Deposition of sediment in a turbulent stream. Izd. AN SSSR, Moscow (in Russian). Schlee, J., 1966. A modified Woods Hole Rapid Sediment Analyzer. J. Sediment. Petrol. 36, 403–413. White, S.J., 1970. Plane bed threshold of fine-grained sediments. Nature 228, 152–153. Willetts, B.B., Rice, M.A., Swaine, S.E., 1982. Shape effects in aeolian grain transport. Sedimentology 29, 409–417. Winkelmolen, A.M., 1982. Critical remarks on grain parameters, with special emphasis on shape. Sedimentology 29, 255–265. Yalin, M.S., Karahan, E., 1979. Inception of sediment transport. J. Hydraul. Div., Proc. Am. Soc. Civ. Eng. 105, 1433–1443. Zegzhda, A.P., 1934. Settlement of sand gravel particles in still water. Izd. NIIG 12, Moscow (in Russian).
Sedimentary Geology 119 (1998) 17–23
Entrainment threshold of natural grains in liquids determined empirically from dimensionless settling velocities and other measures of grain size J.P. Le Roux * Department of Geology, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa Received 18 March 1997; accepted 28 January 1998
Abstract Grain size plays a crucial role in sediment transport, but different methods of size analysis do not yield comparable results. Particle diameters obtained by settling tube, for example, cannot be employed in equations based on sieve sizes. Although the majority of experiments on the initiation of sediment motion has been based on sieve data, settling tube analysis is probably superior as both the settling rate and the entrainment threshold of grains are determined by the same factors of size, shape and density. In this paper, empirical equations are provided which allow the determination of a non-dimensional critical shear stress directly from the equivalent sedimentation diameter. The equations apply to the entrainment of well sorted grains with a high sphericity on flat sediment beds, and are valid for particles and liquids of different densities. 1998 Elsevier Science B.V. All rights reserved. Keywords: Reynolds number; entrainment threshold; settling velocity
1. Introduction The initiation of sediment motion is an important concept in many engineering and sedimentological studies. As a result, much work has been conducted to determine the critical shear stresses required to entrain particles of different sizes, shapes and densities in unidirectional currents (e.g. Miller et al., 1977; Miller and Komar, 1977; Yalin and Karahan, 1979; Le Roux, 1991; Bridge and Bennett, 1992). In the theoretical–empirical curves and equations derived from these studies, grain size plays a pivotal role, either being plotted directly against the critical shear Ł Fax:
C27 (1) 808-4336; E-mail: [email protected]
stress or some related parameter, or as part of a dimensionless relationship such as the grain Reynolds number Re* . However, as pointed out by Pettijohn (1975) and Winkelmolen (1982), ‘size’ is an elusive concept, because it can be expressed variously as volume, weight, surface or cross-sectional area, intercepts through particles or projections, and settling velocity. Grain sizes are commonly determined by sieve analysis and consequently expressed in terms of ‘sieve diameters’, which are construed as equivalent spheres. The actual dimensions, however, depend largely on the shape of the grains and can deviate substantially from this measure. The problem is compounded by screen imperfections and differences in sieving procedures (Komar and Cui, 1984).
0037-0738/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 7 - 0 7 3 8 ( 9 8 ) 0 0 0 2 2 - 0
18
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
Similar constraints apply to grain-size analysis by settling tube, where the dimensions are interpreted in terms of ‘equivalent sedimentation diameters’, i.e. the dimensions of spheres with the same settling rates. It is clear that the actual effects of size and shape cannot be evaluated properly without physically measuring the orthogonal axis dimensions of individual grains, but this procedure is extremely time-consuming and impractical when dealing with numerous small particles. Given a choice between sieve and settling tube analysis as a means of determining the grain size, however, one could reason that the equivalent sedimentation diameter, being a behavioural measure incorporating the properties of size, shape and density, would be preferable in entrainment studies where these factors also play a crucial role. Kench and McLean (1997) concluded that grain-size parameters obtained by sieve analysis can grossly distort the interpretation of depositional and energy processes operating within bioclastic environments, and that hydraulic-size estimates provide a much better tool. Willetts et al. (1982) also argued for adopting settling velocity in all investigations of this kind. Unfortunately, the vast majority of studies on sediment transport is based on sieve-size data, so that the latter require some transformation to conform to settling tube grain dimensions. This relationship is explored below, in order to find a simple means of predicting the entrainment threshold of natural grains directly from their measured settling velocity. 2. Previous work Attempts to relate settling velocity to sediment transport have been made in the past, notably by Collins and Rigler (1982); Komar and Clemens (1985), and Bridge and Bennett (1992). Collins and Rigler (op. cit.) conducted experiments on ilmenite, zircon, rutile and cassiterite grains, used in conjunction with comparable quartz data from White (1970). They concluded that the measured grain settling velocity .Wm / is a good indicator of the critical shear stress .−c / of a wide range of densities under hydraulically smooth boundary conditions, for which the following relationship applies: −c D 1:24 Wm0:33
(1)
Eq. 1 is valid only for water at 20ºC and Re* numbers less than 5. Komar and Clemens (1985) also noted that curves of the threshold friction velocity UcŁ versus Wm for grains of different densities converge in the range of Re* numbers studied by Collins and Rigler (1982), which explains their success in deriving Eq. 1 for minerals of contrasting densities. As Eq. 1 is of limited application, therefore, Komar and Clemens (1985) proposed a universal equation valid for a wide range of densities and liquid compositions: ½0:282 ¼ ²s ² Ł g Wm0:154 (2) Uc D 0:482 ² ² where ² s and ² are the densities of the grain and fluid, respectively, g is the gravity constant, and ¼ is the dynamic (molecular) viscosity. From UcŁ ; the critical shear stress − c can be obtained: −c D .UcŁ /2 ²
(3)
Unfortunately, Eq. 2 is also of limited use, as it is valid only for very small grains settling within the Stokes range .Re < 0:5/: Bridge and Bennett (1992) followed a more theoretical approach to model the entrainment of sediment grains, using the settling velocity .W /; the ratio between the local fluid velocity at the level of effective fluid thrust on bed load grains and the shear velocity .a/; the static friction coefficient .Þc /; the bed slope .S/ and a measure of vertical turbulence asymmetry .b/ to define the following threshold equation: s tan Þc cos S sin S (4) UcŁ D W ac2 C b2 tan Þc For hydraulically rough flows, Eq. 4 gives UcŁ ³ 0:1 W (Bridge and Bennett, 1992). The use of Eq. 4, however, requires input from several parameters which are not easily obtained, whereas the simplified relationship between UcŁ and W is highly inaccurate when compared with experimental results (Table 1). The need for a practical, universal equation or set of equations applicable to different fluids and grain densities over a wide range of Reynolds numbers is therefore evident.
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
19
Table 1 Comparison of critical shear stress determined for different quartz grain sizes in water at 20ºC, using existing formulae and equations proposed in this paper D (cm)
0.0031 0.0063 0.0125 0.0250 0.0500 0.1000 0.2000 0.4000 0.8000
Wm
Wds
0.1001 0.4017 1.3338 3.5955 7.6953 14.574 23.025 33.779 43.356
0.0339 0.1369 0.4619 1.2938 2.9577 6.2012 10.960 18.251 25.908
Wdm
0.0395 0.1585 0.5263 1.4187 3.0364 5.7507 9.0851 13.329 17.107
−c Collins and Rigler (1982)
Komar and Clemens (1985)
Bridge and Bennett (1992)
This study
0.5802 0.9177 1.3636 1.8916 2.4315 3.0020 3.4910 3.9616 4.3018
0.5505 0.8446 1.2223 1.6590 2.0971 2.5530 2.9391 3.3074 3.5717
0.0001 0.0016 0.0178 0.1290 0.5911 2.1203 5.2920 11.390 18.764
0.8492 1.2710 1.7423 2.1964 2.8541 6.6074 14.503 29.138 58.275
3. Present approach
way: r ²s ² D Dd D D 3 ²g ¼2
s
ReŁ2
In order to accommodate grains of different densities in various fluids, it is necessary to use a non-dimensional settling velocity, which is given by: s ²2 3 Wd D W (5) ¼g.²s ²/ Similarly, a dimensionless critical shear stress is required, for which the ‘Shields parameter’ is commonly employed. It is expressed as:
Using Eq. 8, the dimensionless grain size Dd for every data point on the Yalin and Karahan (1979) diagram can be determined. From these values of Dd ; the dimensionless settling velocity of sieve-size spheres can be calculated by employing the set of equations given by Le Roux (1992):
D −c =.²s
Wd D .0:2354Dd /2 forDd < 1:2538
²/g D
(6)
where D is the grain size (usually expressed in terms of the sieve diameter). Although there is a considerable data base on threshold values in the literature, only those experiments dealing with non-cohesive, well sorted grains with a relatively high sphericity entrained on flat sediment beds are considered in this study. Most of the data satisfying these requirements were collected by Miller et al. (1977) and Yalin and Karahan (1979). The latter authors included all of the data of Miller et al. (1977), with the exception of Paintal (1971) and Everts (1973). Yalin and Karahan (1979) plotted threshold data from 19 different sources in a diagram of the Shields parameter against the grain Reynolds number Re* , which is given by: ReŁ D U Ł D²=¼
(7)
Le Roux (1991) showed that Re and are related to the non-dimensional grain size Dd in the following *
Wd D .0:208Dd
3
(10)
0:37/
for 2:9074 < Dd < 22:9866 Wd D .0:8255Dd
(9)
0:0652/3=2
for 1:2538 < Dd < 2:9074 Wd D .0:2636Dd
(8)
(11)
5:4/2=3
for 22:9866 < Dd < 134:9215
(12)
Wd D .2:531Dd C 160/1=2 for 134:9215 < Dd < 1750
(13)
In Fig. 1, the data of Yalin and Karahan (1979) have been recast in a plot of against Wd using Eqs. 8–13. Also included in this diagram are the results of Paintal (1971); Everts (1973), and the data for turbulent flow of Govers (1987). For comparison,
20 J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23 Fig. 1. Plot of dimensionless critical shear stress . / against dimensionless settling velocity .Wds / of equivalent spheres based on sieve diameter. Data from Paintal (1971); Everts (1973); Yalin and Karahan (1979), and Govers (1987). The data points for hydraulically smooth conditions of Collins and Rigler (1982) are also shown for comparison.
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
the data for hydrodynamically smooth conditions of Collins and Rigler (1982) are also plotted, although they were not taken into account for the regression line. These plots correlate reasonably well with the rest of the data for turbulent conditions, but cluster below the regression line in Fig. 1. Linear regression of against Wd yields the equations: D 0:0717 log10 Wds C 0:0625 for Wds < 2:5
(14)
D 0:0171 log10 Wds C 0:0272 for 2:5 < Wds < 11
(15)
D 0:045forWds > 11
(16)
21
4. Relationship between settling velocity of sieve-diameter spheres and natural grains As Eqs. 14–16 apply to the hydrodynamic settling velocity of sieve-diameter spheres .Ws /; the relationship between this measure and the actual measured settling velocity of natural grains .Wm / in water must be established. Experimental data of Zegzhda (1934); Arkhangel’skii (1935); Sarskisyan (1958); Kennedy and Koh (1961); Mamak (1964); Schlee (1966); Sanford and Swift (1971); Baba and Komar (1981), and Raudkivi (1990), related to quarter phi sieve sizes and recast in dimensionless form, are plotted in Fig. 2. A polynomial curve fitted to these data yields a correlation coefficient of 0.9852 and shows that the dimensionless settling velocity of sieve-size spheres .Wds / is related to the dimensionless settling velocity of natural grains .Wdm / in the
Fig. 2. Plot of dimensionless measured settling velocity .Wdm / against dimensionless settling velocity of a sphere .Wds / calculated with Eqs. 9–13. Mean values for measured settling velocity from Zegzhda (1934); Arkhangel’skii (1935); Sarskisyan (1958); Kennedy and Koh (1961, table 2), Mamak (1964); Schlee (1966); Sanford and Swift (1971); Baba and Komar (1981, fig. 5), and Raudkivi (1990). In all cases grain sizes were taken as the retaining quarter phi sieve size.
22
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23
following way: 2 Wds D 0:0384 Wdm C 0:8575 Wdm
(17)
Eqs. 14–16 and Eq. 17 can now be combined so that the dimensionless critical shear stress can be calculated directly from the measured dimensionless settling velocities: D
0:0321 ln Wdm C 0:0655
for Wdm < 2:6103
Acknowledgements
(18)
I wish to thank Drs. J.J. Williams, Z. Jiang and an anonymous reviewer for their critical comments, which assisted in improving this paper.
(19)
References
D 0:0087 ln Wdm C 0:0256 for 2:6103 < Wdm < 9:1108
whereas Eqs. 18 and 19 apply to settling tube data. Where grain axis dimensions are available, settling velocities can also be predicted fairly accurately using the equations of Le Roux (1996), allowing direct calculation of the entrainment thresholds.
For dimensionless settling velocities above 9.1108, is constant at 0.045. From ; the actual critical shear stress −c can be calculated using Eq. 6. The value of D is obtained by converting Wm to the dimensionless settling velocity of its corresponding sieve size (Eqs. 5 and 17), and using the inverse of Eqs. 9–13 (see Le Roux, 1992) to obtain the dimensionless grain size Dd : D is then calculated from Dd using Eq. 8. 5. Conclusions Many laboratories routinely use settling tubes to determine sediment grain sizes, which can differ significantly from that obtained by conventional sieve analysis. For example, if the equivalent sedimentation diameter (0.3654 cm) for the last record in Table 1 were substituted for the actual sieve size (0.8 cm), a critical shear stress of 26.6 instead of 58.3 g cm 1 s 2 would be obtained. Settling tube data therefore cannot be used in equations based on sieve sizes to determine entrainment thresholds. Existing equations based on settling velocity (Collins and Rigler, 1982; Komar and Clemens, 1985; Bridge and Bennett, 1992) apply only to very limited conditions and appear to be very inaccurate when compared with a wider range of experimental data (Table 1). The equations provided here are valid for grains of any density in any liquid. (For such grains in air, special conditions apply — see Le Roux, 1997.) They can also be used to determine the dimensionless critical shear stress for Re* values between 0.03 and more than 1000. For sieve data, Eqs. 14–16 can be employed (using the sieve diameter for D/;
Arkhangel’skii, B., 1935. Experimental study of hydraulic coarseness scale of particles. Izv. NIIG 15, Moscow (in Russian). Baba, J., Komar, P.D., 1981. Measurements and analysis of settling velocities of natural quartz sand grains. J. Sediment. Petrol. 51, 631–640. Bridge, J.S., Bennett, S.J., 1992. A model for the entrainment and transport of sediment grains of mixed sizes, shapes, and densities. Water Resour. Res. 28, 337–363. Collins, M.B., Rigler, J.K., 1982. The use of settling velocity in defining the initiation of motion of heavy mineral grains, under unidirectional flow. Sedimentology 29, 419–426. Everts, C.H., 1973. Particle overpassing on flat granular boundaries. Proc. Am. Soc. Civ. Eng. 99, 425–438. Govers, G., 1987. Initiation of motion in overland flow. Sedimentology 34, 1157–1164. Kench, P.S., McLean, R.F., 1997. A comparison of settling and sieve techniques for the analysis of bioclastic sediments. Sediment. Geol. 109, 111–119. Kennedy, J.F., Koh, R.C., 1961. The relation between the frequency distributions of sieve diameters and fall velocities of sediment particles. J. Geophys. Res. 66, 4233–4246. Komar, P.D., Clemens, K.E., 1985. The relationship between a grain’s settling velocity and threshold of motion under unidirectional currents. J. Sediment. Petrol. 56, 258–266. Komar, P.D., Cui, B., 1984. The analysis of grain-size measurements by sieving and settling-tube techniques. J. Sediment. Petrol. 54, 603–614. Le Roux, J.P., 1991. A rapid method to determine the critical shear stress for sphere entrainment under unidirectional fluid flow. Sediment. Geol. 75, 1–3. Le Roux, J.P., 1992. Settling velocity of spheres: a new approach. Sediment. Geol. 81, 11–16. Le Roux, J.P., 1996. Settling velocity of ellipsoidal grains as related to shape entropy. Sediment. Geol. 101, 15–20. Le Roux, J.P., 1997. Relationship between aerodynamic entrainment threshold and hydrodynamic settling velocity of particles. Sediment. Geol. 109, 199–205. Mamak, W., 1964. River regulation. Translated from Polish by U.S. Dept. of Interior and NSF. Washington, DC. Miller, M.C., Komar, P.D., 1977. The development of sediment
J.P. Le Roux / Sedimentary Geology 119 (1998) 17–23 threshold curves for unusual environments (Mars) and for inadequately studied materials (foram sands). Sedimentology 24, 709–721. Miller, M.C., McCave, I.N., Komar, P.D., 1977. Threshold of sediment motion under unidirectional currents. Sedimentology 24, 507–527. Paintal, A.S., 1971. A stochastic model for bed load transport. J. Hydraul. Res. 9, 527–553. Pettijohn, F.J., 1975. Sedimentary Rocks (3rd ed.). Harper and Row, New York, 628 pp. Raudkivi, A.J., 1990. Loose Boundary Hydraulics. Pergamon Press, New York. Sanford, R.B., Swift, D.J.P., 1971. Comparison of sieving and settling techniques for size analysis, using a Benthos Rapid Sediment Analyzer. Sedimentology 17, 257–264.
23
Sarskisyan, A.A., 1958. Deposition of sediment in a turbulent stream. Izd. AN SSSR, Moscow (in Russian). Schlee, J., 1966. A modified Woods Hole Rapid Sediment Analyzer. J. Sediment. Petrol. 36, 403–413. White, S.J., 1970. Plane bed threshold of fine-grained sediments. Nature 228, 152–153. Willetts, B.B., Rice, M.A., Swaine, S.E., 1982. Shape effects in aeolian grain transport. Sedimentology 29, 409–417. Winkelmolen, A.M., 1982. Critical remarks on grain parameters, with special emphasis on shape. Sedimentology 29, 255–265. Yalin, M.S., Karahan, E., 1979. Inception of sediment transport. J. Hydraul. Div., Proc. Am. Soc. Civ. Eng. 105, 1433–1443. Zegzhda, A.P., 1934. Settlement of sand gravel particles in still water. Izd. NIIG 12, Moscow (in Russian).