Relationship between aerodynamic entrainment threshold and hydrodynamic settling velocity of particles

Sedimentary Geology ELSEVIER

Sedimentary Geology 109 (1997) 199-205

Relationship between aerodynamic entrainment threshold and hydrodynamic settling velocity of particles J.R le Roux Department of Geology University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa

Received 8 March 1996; accepted 19 June 1996

Abstract

Data on the critical shear stress (rc) at incipient grain motion under air flow have been carefully selected from the literature. It is shown that the non-dimensional Shields entrainment threshold (tic) in air is an exponential function of the dimensionless settling velocity (Wds) in water, of a sphere with a diameter equal to the sieve size. As Was can be related to the measured dimensionless settling velocity (Wdm) of natural grains, Wdm can be used directly to calculate/~c. For quartz grains in air, the equations are valid for sizes between 0.01 and at least 0.166 cm, i.e. silt to very coarse sand. The method is applicable to well sorted grains with a relatively high sphericity entrained on fiat surfaces. Keywords: Aeolian transport; Entrainment threshold; Settling velocity

1. Introduction In studies of sediment transport by wind over the last decade, the focus has been mainly on the effect of grain dislodgement caused by the ballistic impact of bouncing grains (e.g. Rumpel, 1985; Willetts and Rice, 1986; Anderson, 1986, 1987; Gillette and Stockton, 1986; Mitha et al., 1986; Werner and Haft, 1988; Anderson and Haft, 1988; Willetts et al., 1991; McEwan et al., 1992; Li and Martz, 1995). However, most experiments indicate that dynamic grain dislodgement in air begins almost immediately after the entrainment of the first few grains, so that the impact threshold may fall within the scatter of aerodynamic thresholds produced by variables such as the shape and degree of exposure of individual grains. Iversen et al. (1987), for example, noted that the increase in the threshold friction or critical shear velocity (U.c) from the first oscillating motion to

full continuous saltation in air is nearly negligible for particle-to-fluid density ratios (Ps/pf) of about 900 to 9000. The ability to predict aerodynamic thresholds therefore, has an important bearing on all aeolian transport studies, but has received relatively little attention since the earlier investigations of Bagnold (1941), Zingg (1953), Chepil (1959) and Miller et al. (1977). Williams et al. (1994), for example, in a paper examining the influence of boundary layer flow conditions on aerodynamic entrainment thresholds, stated that "accurate prognostic determination of threshold conditions is still not possible". The aerodynamic entrainment threshold equation of Bagnold (1941) is still one of the most widely used today. U.c = A ~ / [ g D ( p s - pf)/pf]

(1)

where g is the gravity constant (about 980 cm/s2), D is the particle diameter, and A is an empirical

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J.P. le Roux/Sedimentary Geology 109 (1997j 199-205

coefficient equal to 0.1 for particle friction Reynolds numbers (Re,) > 3.5. Iversen and White (1982) provided two threshold equations for Re, numbers of 0.03-10, and >10, respectively: ~ c c = 0.129 ~/(1 + O'O06/psgD2S)

(2)

V/(1.928 Re °°92 - 1)

x [I - 0.0858 exp-°'°617 IRe.-,o,]

(3)

where/3c is the non-dimensional critical shear stress of Shields (1936). The usefulness of these equations is limited, unfortunately, by the fact that Re. must be known, which requires measuring U.c since: Re, = pfDU, c/I.t

(4)

/~ being the dynamic viscosity of the fluid. Finally, lversen et al. (1987) proposed an empirical expression for Re, numbers > l0 and particle diameters > 0.02 cm:

~CC 0.2 v/{l + 2.311 - exp(-O.0054((p.~/pf) - l)°s6)]} (5) It is clear that existing aerodynamic threshold equations are either restricted to specific Re, numbers and grain sizes, or else require determination of the critical shear stress in any case. The need for an accurate equation or set of equations which can be used to predict the critical boundary shear stress of particles directly from their size and density, as well as the density and viscosity of air at different temperatures, is thus apparent. In this paper, an exponential relationship between the dimensionless settling velocity of sieve-size spheres (Wd.0 in water and /4c is demonstrated. As the fall velocity of well sorted sediments can be determined easily by settling tube and can even be predicted with a fair degree of accuracy for spheres and ellipsoids (Le Roux, 1992, 1996), calculation of the entrainment threshold ~'c is facilitated.

2. Data sources The transport threshold of particles in wind is influenced by a number of factors, including the air temperature, bed geometry, the size, density and shape of the particles, as well as the sorting and packing of the sediment. As these variables can produce an unmanageable amount of scatter, only experiments pertaining to well sorted, relatively high-sphericity grains entrained on flat beds were considered in this study. Data sources satisfying these criteria include Zingg (1953), Chepil (1959), Greeley et al. (1974), Iversen and White (1982), and Nickling (1988). The density of sediments used in these experiments varied between 0.21 and 11.35 g cm -3 (210-11,350 kg m-3). Some of the authors (e.g. Iversen and White, 1982) also included data on material such as crushed walnut shells, but these were excluded from the present study due to the angularity of the particles. Incipient grain motion in all the experiments was determined by means of laser beam optical systems, with the exception of Zingg (1953) and Chepil (1959). However, in spite of the limitations of visual observation, these earlier data sets proved consistent with those produced by more sophisticated methods. Cohesion between grains, which arises mainly from the effects of moisture, Van der Waal's forces and electrostatic charges, can also dramatically increase the critical shear stress of particles smaller than about 0.01 cm in air. For these size ranges, the low-pressure wind-tunnel data for 0.0154 cm sand particles of Iversen and White (1982) were employed. Changing the air pressure in this case simulated threshold conditions for cohesionless quartz grains between 0.0066 cm and 0.0012 cm in diameter at one atmosphere. 3. A p p r o a c h and data analysis The selection criteria employed in this study accentuate the effects of particle size (D) and density (p~), as well as the physical properties of the air (density Of and dynamic viscosity /.z), on entrainment thresholds. It is therefore convenient to approach the subject using a behavioural measure such as the settling velocity (W), which encompasses all of the attributes mentioned above. This also eliminates some of the considerable difficulties

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J.P. le Roux/Sedimentary Geology 109 (1997) 199-205

Chepil (1959) and Zingg (1953) as /9c against Re,, whereas Williams et al. (1994) plotted/3¢ against the Yalin (1977) parameter, which is given by Re2,//9¢. Le Roux (1991) demonstrated that the third root of the Yalin parameter is a dimensionless grain size Dd:

associated with the definition and determination of 'grain size'. (For a discussion of these problems, see Winkelmolen, 1982.) Collins and Rigler (1982), as well as Komar and Clemens (1985) pointed out that the settling velocity of particles in water can be used to determine the critical shear stress for grains of a wide density range. Unfortunately, their threshold equations apply only to very low Reynolds numbers and low Ps/Pf ratios, so that they are not valid for gases such as air. In the approach followed here, a dimensionless settling velocity (Wd) is employed, which is given by: Wd

W ~/{P2/[lzg(Ps

:

-

Pf)]}

Od = O~/[pfg(ps - pf)//z 2] ---- ~ / / 9 c )

Where important variables such as grain sizes are not provided directly in the literature, the relevant information can be obtained from these relationships. The dimensionless grain sizes thus obtained were used to calculate the dimensionless hydrodynamic settling velocities of spheres with diameters equal to Ds, using the equations of Le Roux (1992):

(6)

Wd = (0.2354Da) 2

For a non-dimensional expression of the entrainment threshold, the critical shear stress function/9c of Shields (1936) is convenient. It is given by: /9¢ = rc/(Ps -- pf)gD

W d

-

(0.208Dd

=

for

for

Dd < 1.2538

(10)

0.0652) 3/2

--

1.2538 < Dd < 2.9074

(11)

(7) Wd = (0.2636Dd -- 0.37)

In the majority of threshold experiments reported in the literature, grain sizes were determined by sieve analysis, so that D is equal to the sieve diameter D~. /9¢ is related to the critical shear velocity U,¢ in the following way: U.c = ,J[/gcgD(ps

(9)

Pf)/Pd

for

2.9074 < Da < 22.9866

Wd = (0.8255Dd for

(8)

(12)

5.4) 2/3

-

22.9866 < Dd < 134.9215

(13)

The dimensionless settling velocities were then plotted against the dimensionless aerodynamic critical shear stress. As shown in Fig. 1, a linear relationship is defined when a logarithmic scale is employed

Information on entrainment thresholds in air is reported in the literature in various ways. Miller et al. (1977) plotted the data of Greeley et al. (1974),

.c

o

0.02

_= = o

-0.01

g E Q

0.01 |

l

,

i

I

0.1 i

i

i

i

i

i

I

i

I

1.0 I

I

I

i

I

I

i

i

I

10 i

i

i

l

•

,

,

.

I

Dimensionless settling velocity

Fig. 1. Plot of non-dimensional critical shear stress #¢ of particles in wind against dimensionless hydrodynamic settling velocity of sieve-size spheres Wds.Data from Zingg (1953), Chepil (1959), Greeleyet al. (1974), Iversenand White (1982), and Nickling (1988).

.LP. le Rou.r/Sedimentao" Geology 109 (1997) 199-205

202

.11

f o r Wds"

/5c = -0.00741 logt0 Wds + 0.01495 for

-10

(14)

0.004 < Wd~ < 2.5

0~

E

"9

tic = 0.00664 logl0 Wd, + 0.00936 for

2.5 < Wds < 10

(15)

Two aspects should be kept in mind with regard to Fig. 1. Firstly, the grain sizes are based on sieve analyses, and Eqs. (14) and (15) therefore cannot be used to calculate entrainment thresholds from measured settling rates. Secondly, Eqs. (10)-(13) apply to the fall velocities of perfectly spherical grains, and are thus not directly comparable to the settling rates of the natural grains used in the data base. To accommodate these discrepancies, an empirical expression of the relationship between the measured settling velocities (Win) of natural grains, and the calculated settling velocities (W,) of spheres with the same diameter as the sieve size (as used in Fig. 1), is required. This relationship was examined for natural sandsized particles in two independent studies by Kennedy and Koh (1961) and Baba and Komar ( 1981). The latter authors used the equation of Gibbs et al. ( 197 I) to predict the settling velocities of equivalent spheres (W,) directly from the sieve diameter, which they plotted against Win. In Fig. 2, the average measured settling velocity for each sieve size fraction as obtained by Kennedy and Koh (1961, table 2) and Baba and Komar (1981, fig. 5) was plotted against the predicted settling velocity of sieve-size spheres. In this case, however, Eqs. (10)-(13) were employed, as these are more accurate than the equations of Gibbs et al. (1971) or Dietrich (1982). The settling velocities in Fig. 2 have been recast in dimensionless form using Eq. (6). Linear regression of the plots yields the equations: Wd~ = 0.9143Wdm Wds = 1.4Wdm- 1.7

for for

Wdm < 3.5

(16)

3.5 < Wdm < 8(?) (17)

The data of Kennedy and Koh (1961) and Baba and Komar (198 I) are clearly compatible, especially in the dimensionless measured settling velocity range below 3.5. This section of the graph (Eq. 16) also

"~>~ -8 :~

•

7

•

•

u O

°i

.5

o

o.

-3

•

c

•

•

Kennedy

•

Baba

&

&

Koh

Koma¢

(19611

(19Slb)

E C~

4 t¢

I

I

I

Dimensionless

1

measured

I

settling

I

i

velocity

Fig. 2. Plot of dimensionless measured settling velocity (Wdm) of natural grains against non-dimensional settling velocity of sieve-size spheres (Wd~) calculated with Fqs. (10)-(13). Mean values for measured settling velocities from table 2 in Kennedy and Koh (1961) and fig. 5 in Baba and Komar (1981). passes through the origin, as would be expected. For higher Wdm values there is some scatter, but the cross-plots still define a straight line. How far this linear relationship extends beyond the limits of the experimental data ( Wdmof about 8) is uncertain, but there is no reason why that it should not apply to a Wds value of up to 10, which represents the limit of experimental data in Fig. 1. Combining these equations with Eqs. (14) and (15) yields the following direct relationships between the measured settling velocities and the dimensionless critical shear stress/to: tic = -0.00079 logto WdmJr- 0.01235 for

Wdm < 2.7343

(18)

fie = 0.00653 Iogl0 Wdm + 0.00915 for

2.7343

<

Wdm < 3 . 5

(19)

J.P. le Roux/Sedimentary Geology 109 (1997) 199-205 /~c = 0.00873 logl0 for

Wdm+ 0.00795

Dd = (Wd + 0.37)/0.2636

3.5 < Wdm < 8

for

Wd < 0.0864

(21)

w2/3 + 0.0652)/0.208 Dd = ("'d for

for

(20)

In situations where settling tube analysis is not possible, sieve data or grain axis measurements can also be used to predict the critical shear stress, as long as the sediments are well sorted with a relatively high sphericity. For sieve data, the dimensionless hydrodynamic settling velocities of the sieve-size spheres have to be determined. Eq. (9) is used to first calculate the equivalent dimensionless grain size, which is then e m p l o y e d in Eqs. ( 10)-(I 3) to obtain Wd~. Wds is used directly in Eqs. (14) and (15) to calculate/~c. From /~c as determined by sieve or settling tube analysis, Eq. (7) can be used to obtain the actual entrainment threshold rc. Where the grain size has been calculated from the settling velocity, the value o f D required in Eq. (7) is obtained as follows. Firstly, the dimensionless sieve-size settling velocity is determined from Eqs. (6), (16) and (17). To calculate the equivalent dimensionless grain size Dd~, the following equations are used (Le Roux, 1992): Dd = ~ / 0 . 2 3 5 4

203

0.0864 < Wd < 0.3946

(22)

0.3946 < Wd < 5.6899

(23)

Dd = ( W 3/2 + 5.4)/0.8255 for

5.6899 < Wd < 22.3966

(24)

Ds is obtained from Dd using Eq. (9).

4. Comparison with experimental data and other threshold equations In Table 1, the critical shear velocities as determined from sieve data using the method outlined above, are compared with the experimental values of U,c reported by Iversen and W h i t e (1982, table 1) and Nickling (1988, table 1). The grain sizes for the Nickling data were recalculated to the fourth decimal from the given phi-values. Only grains exceeding 0.01 cm in diameter and having a density o f more than 0.9982 are included, as finer grains are affected by cohesion forces and grains less dense than water would float. The experimental and predicted values show a close correspondence. Table 1 also compares the predicted friction velocities using the threshold equations o f Bagnold (1941), Iversen and White (1982) and Iversen et al. (1987), where

Table 1 Comparison of critical shear stress U,c as predicted by different equations, with experimental results reported by Iversen and White (1982) and Nickling (1988) No.

Ps

Ds

U,cobs.

U,c (LR)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1.30 2.50 2.50 2.50 2.50 2.50 2.50 2.50 2.50 3.99 3.99 11.35 2.65 2.65 2.65 2.65

0.1290 0.0123 0.0174 0.0212 0.0252 0.0347 0.0413 0.0488 0.0586 0.0110 0.0519 0.0720 0.0774 0.0514 0.0268 0.0186

43.2 22.8 23.4 25.5 27.5 28.9 30.8 35.5 38.2 24.8 48.2 101.8 49.0 38.0 27.0 23.0

43.3 21. I 23.7 25.4 27.0 30.3 32.2 35.0 39.3 24.7 47.3 101.1 48.2 37.4 28.3 24.9

U,c (B) 37.0 20.8 22.7 26.6 29.0 31.6 34.6 41.1 81.7 40.9 33.3 24.1 -

U*c (IW) 43.5 21.0 23.1 24.5 26.0 29.2 31.3 34.2 38.2 25.2 46.4 96.8 47.5 37.2 28.3 25.0

U-c (IA) 42.4 23.1 25.2 29.5 32.2 35.0 38.4 45.3 89.9 45.3 36.9 26.7 -

p and /z used for water and air: 0.9982 and 0.0012, 0.01002 and 0.00018, respectively. LR = equations proposed in this paper; B = Bagnold equation (194 !); IW = Iversen and White equations (1982); IA = iversen et al. equations (1987).

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J.P. le Rou_t/Sedimentary Geology 109 (1997) 199-205

applicable. In 12 out of 16 samples, the U.c values predicted according to the method proposed in this paper, are the most accurate.

5. Conclusions As illustrated in Table 1, threshold equations employed to date are not always applicable or accurate under a wide range of aerodynamic conditions. The technique proposed here is restricted to the entrainment of cohesionless, well sorted grains with a relatively high sphericity on fiat surfaces, but these conditions are commonly met in aeolian transport. The use of dimensionless relationships indirectly incorporating crucial factors such as Re., has the advantage that the equations are valid for all cases where the particle-to-fluid density ratio is between about 900 and 9500 (cf. sample 12 in Table 1). This means that even extreme air temperatures (-50°C to more than 40°(2) can be accommodated. For grains of quartz density, the equations apply to sizes between about 0.001 cm (0.01 cm if cohesion is a factor) and at least 0.166 cm. The last figure is defined by the upper limit of experimental data used to establish the relationship between sieve sizes and measured settling rates (Fig. 2). If it is assumed that the linear relationship for larger grain sizes in Fig. 2 extends to a dimensionless settling velocity of about 10 (the limit of experimental data in Fig. 1), the equations would still apply to quartz grains up to 0.178 cm in diameter.

Acknowledgements I wish to thank Dr. J.J. Williams, Dr. I. McEwan and an anonymous reviewer for their much-valued comments on the original manuscript.

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