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Modification of the Engelund bed-load formula
Author’s Accepted Manuscript Modification of the Engelund Bed-Load Formula Zhen Meng, Li Dan-xun, Xing-kui Wang
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S1001-6279(16)30022-1 http://dx.doi.org/10.1016/j.ijsrc.2015.09.002 IJSRC71
To appear in: International Journal of Sediment Research Received date: 17 February 2015 Revised date: 4 September 2015 Accepted date: 8 September 2015 Cite this article as: Zhen Meng, Li Dan-xun and Xing-kui Wang, Modification of the Engelund Bed-Load Formula, International Journal of Sediment Research, http://dx.doi.org/10.1016/j.ijsrc.2015.09.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Modification of the Engelund Bed-Load Formula
Zhen MENG1, Dan-xun LI2, and Xing-kui WANG3 1
PH.D. Student, State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084,China. E-mail:[email protected] 2 Prof. Dr., State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084,China. E-mail: [email protected] 3 Prof. Dr., State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084,China. E-mail: [email protected] Correspondence information： Prof. Dan-xun Li State Key Laboratory of Hydroscience and Engineering, Tsinghua University. Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China. Tel：+86 10 62781747; E-mail: [email protected]
Abstract The classic Engelund bed-load formula involves four oversimplified assumptions concerning the quantity of particles per unit bed area that can be potentially entrained into motion, the probability of sediment being entrained into motion at a given instant, the mean velocity of bed-load motion, and the dimensionless incipient shear stress. These four aspects are reexamined in the light of new findings in hydrodynamics, and a modified bed-load formula is then proposed. The modified formula shows promise as being reliable in predicting bed-load transport rates in a wide range of flow intensities. Key words: Bed-load transport, Incipient shear stress, Entrainment probability, Flow intensity
1 Introduction Bed-load transport occurs as the motion of sediment particles along the channel bed by rolling, sliding, and/or saltating (Chien & Wan, 1983). As it plays an important role in a variety of scientific and engineering settings, bed-load transport has remained a key research interest in the hydraulic community since the pioneering work of Duboys (1879). To predict bed-load transport rate, numerous formulas have been proposed by researchers from across the world. Existing bed-load formulas can be generally classified into two categories, i.e., empirical and semi-empirical (Zanke, 2001). Empirical formulas, such as those by Meyer-peter and Müller (1948), Parker (1978), Cheng (2002), and Knack and Shen (2015), are based purely on regressive analysis of -1-
measured data. Semi-empirical formulas, in contrast, involve both theoretical derivations and statistical analysis. Typical semi-empirical formulas include those proposed by Einstein (1950), Bagnold (1973), Yalin (1977), Engelund and Fredsøe (1976), Ashida and Michiue (1972), Fernandez Luque and van Beek (1976), Fredsøe and Deigaard (1992), Van Rijn (1984), Wilson (1987), Soulsby and Damgaard (2005), Lajeunesse et al. (2010), Zhong et al.(2012), and Bialik and Czernuszenko (2013). Popular bed-load formulas, either empirical or semi-empirical, perform generally well within their domains of validity. Outside the domain of validity, however, even a well-established formula may over-predict or under-predict real bed-load transport by several orders of magnitude (Yu et al., 2009; Recking, 2010; Talukdar et al., 2012; Yu et al., 2012). For example, the Einstein (1950) formula fits well with measured data at weak to moderate transport rate, but deviates considerably at high transport rate (Wang et al., 2008). The applicability of classical bed-load formulas can be extended by modifying some assumptions and oversimplifications made in the original derivations in the light of new findings in hydrodynamics. Examples of such extensions include the modification of the Meyer-peter and Müller (1948) formula by Wong and Parker (2006) and Huang (2010), and the modification of the Einstein (1950) formula by Yalin (1977), Sun and Donahue (2000), Wang et al. (2008), and Armanini et al. (2014). In the present study, we try to modify the Engelund formula. The classical formula starts from the universal relationship for sediment flux in volume, qb: qb =
6
D3 Nb Pub
(1)
where D is particle diameter, Nb is the quantity of particles per unit bed area that can be potentially entrained to motion, P is probability of sediment particle to be entrained into motion at any instant, and ub is mean velocity of moving particles in flow direction. The oversimplification of Nb, ub, P, and Θc (Θc is dimensionless incipient shear stress) leads to drawbacks in the Engelund formula. The drawbacks have been pointed out by several researchers, such as Chien and Wan (1983), Fredsøe and Deigaard (1992), Zhang and Mcconnachie (1994), and Qu (1998), but none of these researchers offered satisfactory modification to the original formula. The present paper will modify some assumptions and propose a new version of the classical Engelund bed-load formula.
2 Modification of the Engelund Formula 2.1 Determination of Nb Engelund assumed that the quantity of particles per unit bed area that can be potentially entrained to motion is as follow, Nb =
1 D2
(2)
Such an assumption, while reasonable at low transport intensity, is improper in flows with high shear stress (Zhang & Mcconnachie, 1994). At low shear stress, bed-load particles move roughly in a single layer; when the flow becomes sufficiently powerful, however, bed-load transport becomes multi-layered, or even in sheet, with a thickness of δb larger than particle diameter (Van Rijn, 1984; Wilson, 1987; Wilson, -2-
1989; Nnadi & Wilson, 1992; Sumer et al., 1996; Abrahams, 2003; Seizilles et al., 2014). To correct the original Nb, the following relationship is recommended: b 1
Nb =
(3)
D D2
Considerable progress has been made in quantifying δb. For example, Einstein (1950) assumes b = 2D regardless of flow conditions, Van Rijn (1984) equals δb to the saltation height with a maximum of 10D , and Wilson (1987) proposes b = 10D , where u*2 ( s ) gD is dimensionless shear stress (ρ is the fluid density, ρs is the sediment density, u* is the friction velocity corresponding to skin friction, and g is the gravitational acceleration). Here we introduce a new assumption for "effective" thickness of bed-load layer as follows, b /D = m c
(4)
where m is a constant coefficient. Substituting Eq. (4) into Eq. (3) yields a new version of Nb: N b = m c
1 D2
(5)
2.2 Determination of P In the original Engelund formula, the probability P is determined as follows: P
6
c
(6)
where is the dynamic friction coefficient of submerged sediment particles. The fact that calibration against measured data yields P greater than unity in flows of 0.5 prompted Engelund to revise Eq.(6) to the following form: 6 4 P 1 c
0.25
(7)
Unfortunately, introduction of Eq.(7) leads to a disastrous under-prediction of high transport rates (Chien & Wan, 1983; Zhang & Mcconnachie, 1994). Here we believe that the approach proposed by Einstein (1950) to determine P is both theoretically sound and practically feasible. Einstein (1950) assumed that P corresponds to the possibility of "the dynamic lift on the particle is larger than its submerged weight", which can be determined as follows: P 1
1
B* 1 0
B* 1 0
et dt 2
(8)
where B* = 1/ 7.0 and 0 = 1/ 2 are constant coefficients. With Eqs. (5) and (8) we can calculate the number, nb , of particles in motion per unity bed area at any given instant such that, nb = Nb P = mP c / D2
(9)
Lajeunesse et al. (2010) recently reported experimental results, showing that nb increases linearly with ( c ) as follows, -3-
nb c D2
(10)
where is a varying coefficient. It is seen that Eqs. (9) and (10) are strikingly similar. 2.3 Determination of ub The original Engelund formula used the following expression for mean bed-load velocity,
ub = 9.3u* 1 0.7 c
(11)
Eq. (11) is derived via a simplified average sliding kinetic equilibrium equation, and it follows the general form of bed-load velocity (Hu & Hui, 1996),
ub = u* 1 c
(12)
where and are coefficients. Recently, Lajeunesse et al. (2010) recommended a similar formulation of the form, ub = u*
1 c
(13)
where = 4.4 0.2 and = 0.11 0.03 . Eqs. (11) and (13) are problematic, as they yield nonzero velocities at threshold conditions of = c , contradicting the basic assumption made in the deterministic approach which stipulates that no bed-load transport occurs at c . Here we propose a new expression for ub based on equilibrium analysis involving three major forces that govern the motion of an submerged particle, namely, the drag force FD, the lift force FL, and the submerged weight FG: FD CD FL CL FG
6
2
u
u 2
f
f
ub ub
2
2
4
4
D2 D2
(14)
s gD3
where u f = u* is the flow velocity at (1~2)D above bed, = 6~10 , and CD and CL are drag and lift coefficients, respectively. At equilibrium condition, one gets, FD
FG FL
(15)
Substituting Eq. (14) into Eq. (15) yields, CD
2
u
f
ub
2
4
2 D 2 = s gD3 CL u f ub D 2 2 4 6
(16)
Substituting u f = u* into Eq. (16) yields, s gD 4 ub = u* 1 2 3 CD + CL u*2
(17)
At incipient conditions corresponding to ub = 0 and u* = u*c ( u*c is the incipient friction velocity), Eq. (17) simplifies to
-4-
c
u*2c 4 = 2 gD 3 C s D +CL
(18)
Substituting Eq. (18) into Eq. (17) yields, c ub = u* 1
(19)
2.4 Selection of Θc The original Engelund formula selected c = 0.046 for incipient motion, resulting in negative transport rate for 0.046 where bed-load transport still prevails as evidenced by Einstein (1950), Parker (1978), and Cheng (2002). Several studies reported values for c ranging from 0.03 to 0.06 (White et al., 1975; Lavelle & Mofjeld, 1987; Martin, 2003). Some researchers recommended that an appropriate value for c should be closely related to particle characteristics (Van Rijn, 1984; Buffington and Montgomery, 1997; McEwan and Heald, 2001; Recking, 2009; Crookston and Tullis, 2011), while others argued that the introduction of c is not necessary at all (Einstein, 1950; Lavelle & Mofjeld, 1987). We believe that the concept of dimensionless incipient shear stress is of great importance to bed-load transport prediction. Choosing an appropriate value for c can greatly enhance bed-load transport predictions (Martin, 2003; Recking, 2009; Patel et al., 2013; Bussi et al., 2014). The present paper chooses c = 0.03 for bed-load transport rate prediction, in agreement with Neill (1968) and Parker (1978). 2.5 The modified formula Substituting Eqs. (5), (8), and (19) into Eq. (1) yields the modified version of the Engelund bed-load formula, kP c
where qb
s gD3
c
(20)
is dimensionless bed-load transport rate,
qb
is
bed-load transport rate in volume per unit time and width, and k m 6 is a constant coefficient to be calibrated based on experimental data. Comparison of the modified and the original formulas are listed in Table 1. Table 1 Comparison of the modified and the original formula
*
Nb
P
ub
c
Original
1 D2
6 c
c 9.3u* 1 0.7
0.046
Modified
m c D2
1
1
B* 1 0
B* 1 0
et dt 2
The parameter of m 6( k ) will be given in the following section.
-5-
u* 1
c
0.03
3 Performance evaluation and discussion 3.1 Parameter calibration Not a single bed-load formula performs well consistently without site specific recalibrations (De Sutter et al., 2003). As bed-load measurement involves considerable uncertainty (Engelund & Fredsøe, 1976; Martin, 2003; Gomez, 2006), care needs to be taken when one selects measured date to calibrate models. We select well-recognized experimental data sets for parameter calibration, including those by Gilbert (1914), Meyer-Peter and Müller (1948), Wilson (1966), and Roseberry et al. (2012). These data sets have already been examined in several previous studies (Chien & Wan, 1983; Soulsby & Damgaard, 2005; Nnadi & Wilson, 1992; Zhang & Mcconnachie, 1994; Cheng, 2002; Roseberry et al., 2012; Armanini et al., 2014; Dey, 2014). The selected data sets produces k = 14.2 together with parameters of c = 0.03 , B* = 1/ 7.5 , and 0 = 1/ 2 . 3.2 Comparison with classical formulas Six typical bed-load transport formulas were selected for comparison with the new modified Engelund formula (see Table 2). The results are plotted in Fig. 1 and Fig. 2. Table 2 Typical bed-load formulas of the form = f ()
Authors Meyer-peter and Müller (1948) Einstein (1950)
Notes
= K c
1.5
=
K 8 , c = 0.047
P 1
P 0 A* 1 P
= K
1
c
= K c
Cheng (2002)
= K 1.5 e
Engelund (1976)
= K c
Present
= kP c
4.5
et dt , 2
30.2mD w 1 5.75log , tan kS u* 0.3
tan = 0.63,
D = 1, m = 1.4 , kS c
w = 4.5 u* c
Parker (1978)
B* 1 0
B* 1 0
B* = 1/ 7.0, 0 = 1/ 2, A* = 1/0.023 K=
Bagnold (1973)
= f ()
3
0.5
(for D 0.7mm)
K = 11.2 , c = 0.03
0.05
K = 13
1.5
0.7 c
c
c = 0.046, K = ,
= 9.3, = 0.8 k = 14.2, c = 0.03, B* = 1/ 7.5, 0 = 1/ 2
Fig. 1 shows the comparison of the formulas proposed by Einstein (1950), Bagnold -6-
(1973), Engelund (1976) and the present modified version. It is clear that the prediction by Eq. (20) agrees well with the measured data, and the original Engelund formula yields a smaller transport rate than the modified version. The Einstein formula reports far smaller transport rate in high flows ( 3 ) where the Bagnold formula tends to yield a slight overestimation.
Fig.1 Comparisons with the formulas derived by Einstein (1950), Bagnold (1973), and Engelund (1976).
Fig. 2 shows the comparison of the modified Engelund formula with other formulas proposed by Meyer-Peter and Müller (1948), Parker (1978), and Cheng (2002). One can see the modified formula is very close to the formula by Cheng (2002) and the other two formulas report smaller transport rates.
Fig.2 Comparisons with the formulas derived by Meyer-Peter and Müller (1948), Parker (1978), and Cheng (2002). -7-
3.3 Error analysis Different efficiency criterion can be used to measure the performance of bed-load formulas, such as the indexes of agreement Id (Talukdas et al., 2012), root mean square relative error RMSRE (Wang et al., 2008) and discrepancy ratio R (Haddadchi et al., 2013). The index of agreement Id, which represents the ratio of the mean square error to the potential error, is defined as: n
Id 1
i 1
n
i 1
M i
2
P
P M M M
(21)
2 i
where n is the total number of data used, P and M are predicted and measured values, respectively. The range of Id lies in between zero (no correlation) and unity (perfect fit). The root mean square relative error RMSRE is defined as: 2
RMSRE
1 n P M n i 1 M i
(22)
where zero of RMSRE indicates a perfect fit between predicted and measured values. The discrepancy ratio R, which indicates under or over prediction by the model, is defined as: Ri P M i
(23)
Table 3 shows the comparison of these indicators for selected formulas. From the results one can see that the modified formula provides satisfactory agreement with the measured data. Note that it is far too early to conclude that the modified formula performs better than other formulas unless a much broader set of measured data are used for further checking. Table 3 Indexes of comparison between predicted and measured transport rates
Method
Data in range of discrepancy ratio, R (%) 0.75~1.25 0.5~1.5 0.25~1.75 0.5~2
Id
RMSRE
R
Meyer-peter and Müller (1948)
24.4
87.4
98.5
89.3
71.3
0.912
0.388
Einstein (1950)
26.4
49.2
88.3
51.8
65.4
0.559
0.519
Bagnold (1973)
54.8
88.8
97.5
95.4
108.6 0.969
0.334
Parker (1979)
56.3
89.3
100
89.8
78.0
0.985
0.306
Cheng (2002)
70.6
89.8
97.9
92.4
96.6
0.996
0.292
Engelund (1976)
54.8
88.8
97.5
95.4
85.6
0.983
0.311
Present
72.6
94.4
97.5
97.0
100.3 0.996
0.285
*
Excluding 13 sets of Meyer-Peter and Müller's data as the formulas by Meyer-peter and Müller -8-
(1948), Bagnold (1973), and Engelund (1976) report negative bed-load transport rates.
3.4 Discussion The modified Engelund formula yields better predictions than the original version. The improvement is, apparently, attributed to the appropriate determination of Nb, ub, and P. The merit of the modified formula can be further illustrated by introducing a general form of bed-load transport rate as follows, K x y c 0
c
, z
c
, c
(24)
where K, x, y, z, and λ are coefficients. The original Engelund formula adopted a constant K = 11.6, in disagreement with experimental evidence by Fernanctez Lugue and van Beck (1976) (K = 5.7, 0.087 ), Wong and Parker (2006) (K = 3.97, 0.2 ), and Wilson (1987) (K = 12, 0.8 ). Soulsby and Damgaard (2005) reported that the value of K varies with (K = 8 fits better for smaller whereas K = 12 fits better at larger ). In consistence with these findings, the modified Engelund formula enjoys a varying K(= kP), which increases gradually with the increasing flow intensity. The choice of c 0.03 is equally important for the success of the modified formula. However, simply substituting c 0.03 into other formulas does not guarantee better performance. Fig. 3 reveals that using c 0.03 leads to marked over-prediction of bed-load transport by formulas by Meyer-peter and Müller (1948), Bagnold (1973), and Engelund (1976). Thus, it is important to bear in mind that developing bed-load models must rely on a unified way of determining various parameters. The introduction of c in any bed-load formula will inevitably yields negative transport rate at extremely weak flow intensities of c . In concept, this constitutes one inherent drawback of the deterministic approach. In practice, however, using a very small c = 0.03 is sufficient for observing and calculating bed-load transport rate of engineering significance (Lavelle & Mofjeld, 1987).
-9-
Fig.3 Performances of various formulas with
c =0.030
4 Conclusions A modified version of the classic Engelund formula has been proposed. The modification involves four aspects, namely, the quantity of particles per unit bed area that can be potentially entrained into motion (Nb), the probability of sediment being entrained into motion at any instant (P), the mean velocity of bed-load particles (ub), and the dimensionless incipient shear stress ( c ). In the modified formula, Nb is corrected by introducing an “effective” thickness of bed-load layer, P is calculated by following Einstein’s approach, ub reduces to zero at incipient motion, and c = 0.03 is adopted to account for incipient motion instead of the original 0.046. Verification against well-recognized measured data shows that the modified formula provides satisfactory predictions, capable of estimating bed-load transport rate at both low and very strong flow intensities. As bed-load transport formulas rely heavily on data calibration, the applicability of the modified formula needs further verification.
Acknowledgement The research is financially supported by National Natural Science Foundation of China (Grant No.51279081 and 51279012) and National Key Technologies R&D Program of China (Grant No. 2012BAB04B01). The authors are grateful to the anonymous reviewers for their valuable suggestions.
-10-
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APPENDIX : Notation The following symbols are used in this paper: D = diameter of particle (L); -2 g = gravitational acceleration (LT ); 2 -1 qb = bed-load transport rate in volume per unit width (L T ); = dimensionless transport rate; K = modified coefficient of -1 u* = friction velocity corresponding to skin friction (LT ); -1
u*c = incipient friction velocity (LT );
uf = flow velocity near bed (LT -1); -1 ub = mean velocity of bed-load (LT ); x = constant coefficient; y = constant coefficient;
z = constant coefficient;
= constant coefficient; = constant coefficient; m = constant coefficient; k = constant coefficient; = a varied number;
= a coefficient; = constant coefficient; P = sediment entrainment probability; A* = constant coefficient; B* = constant coefficient; 0 = constant coefficient; = dynamic friction coefficient of submerged sediment particles; -3
= density of fluid (ML ); -3
s = density of sediment (ML );
= dimensionless shear stress; c =dimensionless incipient shear stress;
b = thickness of bed-load layer (L); N b = quantity of particles per unit bed area that can be potentially entrained into motion; nb = quantity of moving sediment particles per unit bed area; CD = drag coefficient; CL = lift coefficient; -2 FD = fluid drag force (MLT ); -2 FL = fluid lift force (MLT ); -2 FG = submerged weight (MLT ); -1
w = particle fall velocity of bed material (LT ); k s = equivalent roughness of Nikuradse (L); I d = index of agreement;
RMSRE = root mean square relative error; and
R = discrepancy ratio. -14-
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S1001-6279(16)30022-1 http://dx.doi.org/10.1016/j.ijsrc.2015.09.002 IJSRC71
To appear in: International Journal of Sediment Research Received date: 17 February 2015 Revised date: 4 September 2015 Accepted date: 8 September 2015 Cite this article as: Zhen Meng, Li Dan-xun and Xing-kui Wang, Modification of the Engelund Bed-Load Formula, International Journal of Sediment Research, http://dx.doi.org/10.1016/j.ijsrc.2015.09.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Modification of the Engelund Bed-Load Formula
Zhen MENG1, Dan-xun LI2, and Xing-kui WANG3 1
PH.D. Student, State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084,China. E-mail:[email protected] 2 Prof. Dr., State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084,China. E-mail: [email protected] 3 Prof. Dr., State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084,China. E-mail: [email protected] Correspondence information： Prof. Dan-xun Li State Key Laboratory of Hydroscience and Engineering, Tsinghua University. Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China. Tel：+86 10 62781747; E-mail: [email protected]
Abstract The classic Engelund bed-load formula involves four oversimplified assumptions concerning the quantity of particles per unit bed area that can be potentially entrained into motion, the probability of sediment being entrained into motion at a given instant, the mean velocity of bed-load motion, and the dimensionless incipient shear stress. These four aspects are reexamined in the light of new findings in hydrodynamics, and a modified bed-load formula is then proposed. The modified formula shows promise as being reliable in predicting bed-load transport rates in a wide range of flow intensities. Key words: Bed-load transport, Incipient shear stress, Entrainment probability, Flow intensity
1 Introduction Bed-load transport occurs as the motion of sediment particles along the channel bed by rolling, sliding, and/or saltating (Chien & Wan, 1983). As it plays an important role in a variety of scientific and engineering settings, bed-load transport has remained a key research interest in the hydraulic community since the pioneering work of Duboys (1879). To predict bed-load transport rate, numerous formulas have been proposed by researchers from across the world. Existing bed-load formulas can be generally classified into two categories, i.e., empirical and semi-empirical (Zanke, 2001). Empirical formulas, such as those by Meyer-peter and Müller (1948), Parker (1978), Cheng (2002), and Knack and Shen (2015), are based purely on regressive analysis of -1-
measured data. Semi-empirical formulas, in contrast, involve both theoretical derivations and statistical analysis. Typical semi-empirical formulas include those proposed by Einstein (1950), Bagnold (1973), Yalin (1977), Engelund and Fredsøe (1976), Ashida and Michiue (1972), Fernandez Luque and van Beek (1976), Fredsøe and Deigaard (1992), Van Rijn (1984), Wilson (1987), Soulsby and Damgaard (2005), Lajeunesse et al. (2010), Zhong et al.(2012), and Bialik and Czernuszenko (2013). Popular bed-load formulas, either empirical or semi-empirical, perform generally well within their domains of validity. Outside the domain of validity, however, even a well-established formula may over-predict or under-predict real bed-load transport by several orders of magnitude (Yu et al., 2009; Recking, 2010; Talukdar et al., 2012; Yu et al., 2012). For example, the Einstein (1950) formula fits well with measured data at weak to moderate transport rate, but deviates considerably at high transport rate (Wang et al., 2008). The applicability of classical bed-load formulas can be extended by modifying some assumptions and oversimplifications made in the original derivations in the light of new findings in hydrodynamics. Examples of such extensions include the modification of the Meyer-peter and Müller (1948) formula by Wong and Parker (2006) and Huang (2010), and the modification of the Einstein (1950) formula by Yalin (1977), Sun and Donahue (2000), Wang et al. (2008), and Armanini et al. (2014). In the present study, we try to modify the Engelund formula. The classical formula starts from the universal relationship for sediment flux in volume, qb: qb =
6
D3 Nb Pub
(1)
where D is particle diameter, Nb is the quantity of particles per unit bed area that can be potentially entrained to motion, P is probability of sediment particle to be entrained into motion at any instant, and ub is mean velocity of moving particles in flow direction. The oversimplification of Nb, ub, P, and Θc (Θc is dimensionless incipient shear stress) leads to drawbacks in the Engelund formula. The drawbacks have been pointed out by several researchers, such as Chien and Wan (1983), Fredsøe and Deigaard (1992), Zhang and Mcconnachie (1994), and Qu (1998), but none of these researchers offered satisfactory modification to the original formula. The present paper will modify some assumptions and propose a new version of the classical Engelund bed-load formula.
2 Modification of the Engelund Formula 2.1 Determination of Nb Engelund assumed that the quantity of particles per unit bed area that can be potentially entrained to motion is as follow, Nb =
1 D2
(2)
Such an assumption, while reasonable at low transport intensity, is improper in flows with high shear stress (Zhang & Mcconnachie, 1994). At low shear stress, bed-load particles move roughly in a single layer; when the flow becomes sufficiently powerful, however, bed-load transport becomes multi-layered, or even in sheet, with a thickness of δb larger than particle diameter (Van Rijn, 1984; Wilson, 1987; Wilson, -2-
1989; Nnadi & Wilson, 1992; Sumer et al., 1996; Abrahams, 2003; Seizilles et al., 2014). To correct the original Nb, the following relationship is recommended: b 1
Nb =
(3)
D D2
Considerable progress has been made in quantifying δb. For example, Einstein (1950) assumes b = 2D regardless of flow conditions, Van Rijn (1984) equals δb to the saltation height with a maximum of 10D , and Wilson (1987) proposes b = 10D , where u*2 ( s ) gD is dimensionless shear stress (ρ is the fluid density, ρs is the sediment density, u* is the friction velocity corresponding to skin friction, and g is the gravitational acceleration). Here we introduce a new assumption for "effective" thickness of bed-load layer as follows, b /D = m c
(4)
where m is a constant coefficient. Substituting Eq. (4) into Eq. (3) yields a new version of Nb: N b = m c
1 D2
(5)
2.2 Determination of P In the original Engelund formula, the probability P is determined as follows: P
6
c
(6)
where is the dynamic friction coefficient of submerged sediment particles. The fact that calibration against measured data yields P greater than unity in flows of 0.5 prompted Engelund to revise Eq.(6) to the following form: 6 4 P 1 c
0.25
(7)
Unfortunately, introduction of Eq.(7) leads to a disastrous under-prediction of high transport rates (Chien & Wan, 1983; Zhang & Mcconnachie, 1994). Here we believe that the approach proposed by Einstein (1950) to determine P is both theoretically sound and practically feasible. Einstein (1950) assumed that P corresponds to the possibility of "the dynamic lift on the particle is larger than its submerged weight", which can be determined as follows: P 1
1
B* 1 0
B* 1 0
et dt 2
(8)
where B* = 1/ 7.0 and 0 = 1/ 2 are constant coefficients. With Eqs. (5) and (8) we can calculate the number, nb , of particles in motion per unity bed area at any given instant such that, nb = Nb P = mP c / D2
(9)
Lajeunesse et al. (2010) recently reported experimental results, showing that nb increases linearly with ( c ) as follows, -3-
nb c D2
(10)
where is a varying coefficient. It is seen that Eqs. (9) and (10) are strikingly similar. 2.3 Determination of ub The original Engelund formula used the following expression for mean bed-load velocity,
ub = 9.3u* 1 0.7 c
(11)
Eq. (11) is derived via a simplified average sliding kinetic equilibrium equation, and it follows the general form of bed-load velocity (Hu & Hui, 1996),
ub = u* 1 c
(12)
where and are coefficients. Recently, Lajeunesse et al. (2010) recommended a similar formulation of the form, ub = u*
1 c
(13)
where = 4.4 0.2 and = 0.11 0.03 . Eqs. (11) and (13) are problematic, as they yield nonzero velocities at threshold conditions of = c , contradicting the basic assumption made in the deterministic approach which stipulates that no bed-load transport occurs at c . Here we propose a new expression for ub based on equilibrium analysis involving three major forces that govern the motion of an submerged particle, namely, the drag force FD, the lift force FL, and the submerged weight FG: FD CD FL CL FG
6
2
u
u 2
f
f
ub ub
2
2
4
4
D2 D2
(14)
s gD3
where u f = u* is the flow velocity at (1~2)D above bed, = 6~10 , and CD and CL are drag and lift coefficients, respectively. At equilibrium condition, one gets, FD
FG FL
(15)
Substituting Eq. (14) into Eq. (15) yields, CD
2
u
f
ub
2
4
2 D 2 = s gD3 CL u f ub D 2 2 4 6
(16)
Substituting u f = u* into Eq. (16) yields, s gD 4 ub = u* 1 2 3 CD + CL u*2
(17)
At incipient conditions corresponding to ub = 0 and u* = u*c ( u*c is the incipient friction velocity), Eq. (17) simplifies to
-4-
c
u*2c 4 = 2 gD 3 C s D +CL
(18)
Substituting Eq. (18) into Eq. (17) yields, c ub = u* 1
(19)
2.4 Selection of Θc The original Engelund formula selected c = 0.046 for incipient motion, resulting in negative transport rate for 0.046 where bed-load transport still prevails as evidenced by Einstein (1950), Parker (1978), and Cheng (2002). Several studies reported values for c ranging from 0.03 to 0.06 (White et al., 1975; Lavelle & Mofjeld, 1987; Martin, 2003). Some researchers recommended that an appropriate value for c should be closely related to particle characteristics (Van Rijn, 1984; Buffington and Montgomery, 1997; McEwan and Heald, 2001; Recking, 2009; Crookston and Tullis, 2011), while others argued that the introduction of c is not necessary at all (Einstein, 1950; Lavelle & Mofjeld, 1987). We believe that the concept of dimensionless incipient shear stress is of great importance to bed-load transport prediction. Choosing an appropriate value for c can greatly enhance bed-load transport predictions (Martin, 2003; Recking, 2009; Patel et al., 2013; Bussi et al., 2014). The present paper chooses c = 0.03 for bed-load transport rate prediction, in agreement with Neill (1968) and Parker (1978). 2.5 The modified formula Substituting Eqs. (5), (8), and (19) into Eq. (1) yields the modified version of the Engelund bed-load formula, kP c
where qb
s gD3
c
(20)
is dimensionless bed-load transport rate,
qb
is
bed-load transport rate in volume per unit time and width, and k m 6 is a constant coefficient to be calibrated based on experimental data. Comparison of the modified and the original formulas are listed in Table 1. Table 1 Comparison of the modified and the original formula
*
Nb
P
ub
c
Original
1 D2
6 c
c 9.3u* 1 0.7
0.046
Modified
m c D2
1
1
B* 1 0
B* 1 0
et dt 2
The parameter of m 6( k ) will be given in the following section.
-5-
u* 1
c
0.03
3 Performance evaluation and discussion 3.1 Parameter calibration Not a single bed-load formula performs well consistently without site specific recalibrations (De Sutter et al., 2003). As bed-load measurement involves considerable uncertainty (Engelund & Fredsøe, 1976; Martin, 2003; Gomez, 2006), care needs to be taken when one selects measured date to calibrate models. We select well-recognized experimental data sets for parameter calibration, including those by Gilbert (1914), Meyer-Peter and Müller (1948), Wilson (1966), and Roseberry et al. (2012). These data sets have already been examined in several previous studies (Chien & Wan, 1983; Soulsby & Damgaard, 2005; Nnadi & Wilson, 1992; Zhang & Mcconnachie, 1994; Cheng, 2002; Roseberry et al., 2012; Armanini et al., 2014; Dey, 2014). The selected data sets produces k = 14.2 together with parameters of c = 0.03 , B* = 1/ 7.5 , and 0 = 1/ 2 . 3.2 Comparison with classical formulas Six typical bed-load transport formulas were selected for comparison with the new modified Engelund formula (see Table 2). The results are plotted in Fig. 1 and Fig. 2. Table 2 Typical bed-load formulas of the form = f ()
Authors Meyer-peter and Müller (1948) Einstein (1950)
Notes
= K c
1.5
=
K 8 , c = 0.047
P 1
P 0 A* 1 P
= K
1
c
= K c
Cheng (2002)
= K 1.5 e
Engelund (1976)
= K c
Present
= kP c
4.5
et dt , 2
30.2mD w 1 5.75log , tan kS u* 0.3
tan = 0.63,
D = 1, m = 1.4 , kS c
w = 4.5 u* c
Parker (1978)
B* 1 0
B* 1 0
B* = 1/ 7.0, 0 = 1/ 2, A* = 1/0.023 K=
Bagnold (1973)
= f ()
3
0.5
(for D 0.7mm)
K = 11.2 , c = 0.03
0.05
K = 13
1.5
0.7 c
c
c = 0.046, K = ,
= 9.3, = 0.8 k = 14.2, c = 0.03, B* = 1/ 7.5, 0 = 1/ 2
Fig. 1 shows the comparison of the formulas proposed by Einstein (1950), Bagnold -6-
(1973), Engelund (1976) and the present modified version. It is clear that the prediction by Eq. (20) agrees well with the measured data, and the original Engelund formula yields a smaller transport rate than the modified version. The Einstein formula reports far smaller transport rate in high flows ( 3 ) where the Bagnold formula tends to yield a slight overestimation.
Fig.1 Comparisons with the formulas derived by Einstein (1950), Bagnold (1973), and Engelund (1976).
Fig. 2 shows the comparison of the modified Engelund formula with other formulas proposed by Meyer-Peter and Müller (1948), Parker (1978), and Cheng (2002). One can see the modified formula is very close to the formula by Cheng (2002) and the other two formulas report smaller transport rates.
Fig.2 Comparisons with the formulas derived by Meyer-Peter and Müller (1948), Parker (1978), and Cheng (2002). -7-
3.3 Error analysis Different efficiency criterion can be used to measure the performance of bed-load formulas, such as the indexes of agreement Id (Talukdas et al., 2012), root mean square relative error RMSRE (Wang et al., 2008) and discrepancy ratio R (Haddadchi et al., 2013). The index of agreement Id, which represents the ratio of the mean square error to the potential error, is defined as: n
Id 1
i 1
n
i 1
M i
2
P
P M M M
(21)
2 i
where n is the total number of data used, P and M are predicted and measured values, respectively. The range of Id lies in between zero (no correlation) and unity (perfect fit). The root mean square relative error RMSRE is defined as: 2
RMSRE
1 n P M n i 1 M i
(22)
where zero of RMSRE indicates a perfect fit between predicted and measured values. The discrepancy ratio R, which indicates under or over prediction by the model, is defined as: Ri P M i
(23)
Table 3 shows the comparison of these indicators for selected formulas. From the results one can see that the modified formula provides satisfactory agreement with the measured data. Note that it is far too early to conclude that the modified formula performs better than other formulas unless a much broader set of measured data are used for further checking. Table 3 Indexes of comparison between predicted and measured transport rates
Method
Data in range of discrepancy ratio, R (%) 0.75~1.25 0.5~1.5 0.25~1.75 0.5~2
Id
RMSRE
R
Meyer-peter and Müller (1948)
24.4
87.4
98.5
89.3
71.3
0.912
0.388
Einstein (1950)
26.4
49.2
88.3
51.8
65.4
0.559
0.519
Bagnold (1973)
54.8
88.8
97.5
95.4
108.6 0.969
0.334
Parker (1979)
56.3
89.3
100
89.8
78.0
0.985
0.306
Cheng (2002)
70.6
89.8
97.9
92.4
96.6
0.996
0.292
Engelund (1976)
54.8
88.8
97.5
95.4
85.6
0.983
0.311
Present
72.6
94.4
97.5
97.0
100.3 0.996
0.285
*
Excluding 13 sets of Meyer-Peter and Müller's data as the formulas by Meyer-peter and Müller -8-
(1948), Bagnold (1973), and Engelund (1976) report negative bed-load transport rates.
3.4 Discussion The modified Engelund formula yields better predictions than the original version. The improvement is, apparently, attributed to the appropriate determination of Nb, ub, and P. The merit of the modified formula can be further illustrated by introducing a general form of bed-load transport rate as follows, K x y c 0
c
, z
c
, c
(24)
where K, x, y, z, and λ are coefficients. The original Engelund formula adopted a constant K = 11.6, in disagreement with experimental evidence by Fernanctez Lugue and van Beck (1976) (K = 5.7, 0.087 ), Wong and Parker (2006) (K = 3.97, 0.2 ), and Wilson (1987) (K = 12, 0.8 ). Soulsby and Damgaard (2005) reported that the value of K varies with (K = 8 fits better for smaller whereas K = 12 fits better at larger ). In consistence with these findings, the modified Engelund formula enjoys a varying K(= kP), which increases gradually with the increasing flow intensity. The choice of c 0.03 is equally important for the success of the modified formula. However, simply substituting c 0.03 into other formulas does not guarantee better performance. Fig. 3 reveals that using c 0.03 leads to marked over-prediction of bed-load transport by formulas by Meyer-peter and Müller (1948), Bagnold (1973), and Engelund (1976). Thus, it is important to bear in mind that developing bed-load models must rely on a unified way of determining various parameters. The introduction of c in any bed-load formula will inevitably yields negative transport rate at extremely weak flow intensities of c . In concept, this constitutes one inherent drawback of the deterministic approach. In practice, however, using a very small c = 0.03 is sufficient for observing and calculating bed-load transport rate of engineering significance (Lavelle & Mofjeld, 1987).
-9-
Fig.3 Performances of various formulas with
c =0.030
4 Conclusions A modified version of the classic Engelund formula has been proposed. The modification involves four aspects, namely, the quantity of particles per unit bed area that can be potentially entrained into motion (Nb), the probability of sediment being entrained into motion at any instant (P), the mean velocity of bed-load particles (ub), and the dimensionless incipient shear stress ( c ). In the modified formula, Nb is corrected by introducing an “effective” thickness of bed-load layer, P is calculated by following Einstein’s approach, ub reduces to zero at incipient motion, and c = 0.03 is adopted to account for incipient motion instead of the original 0.046. Verification against well-recognized measured data shows that the modified formula provides satisfactory predictions, capable of estimating bed-load transport rate at both low and very strong flow intensities. As bed-load transport formulas rely heavily on data calibration, the applicability of the modified formula needs further verification.
Acknowledgement The research is financially supported by National Natural Science Foundation of China (Grant No.51279081 and 51279012) and National Key Technologies R&D Program of China (Grant No. 2012BAB04B01). The authors are grateful to the anonymous reviewers for their valuable suggestions.
-10-
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APPENDIX : Notation The following symbols are used in this paper: D = diameter of particle (L); -2 g = gravitational acceleration (LT ); 2 -1 qb = bed-load transport rate in volume per unit width (L T ); = dimensionless transport rate; K = modified coefficient of -1 u* = friction velocity corresponding to skin friction (LT ); -1
u*c = incipient friction velocity (LT );
uf = flow velocity near bed (LT -1); -1 ub = mean velocity of bed-load (LT ); x = constant coefficient; y = constant coefficient;
z = constant coefficient;
= constant coefficient; = constant coefficient; m = constant coefficient; k = constant coefficient; = a varied number;
= a coefficient; = constant coefficient; P = sediment entrainment probability; A* = constant coefficient; B* = constant coefficient; 0 = constant coefficient; = dynamic friction coefficient of submerged sediment particles; -3
= density of fluid (ML ); -3
s = density of sediment (ML );
= dimensionless shear stress; c =dimensionless incipient shear stress;
b = thickness of bed-load layer (L); N b = quantity of particles per unit bed area that can be potentially entrained into motion; nb = quantity of moving sediment particles per unit bed area; CD = drag coefficient; CL = lift coefficient; -2 FD = fluid drag force (MLT ); -2 FL = fluid lift force (MLT ); -2 FG = submerged weight (MLT ); -1
w = particle fall velocity of bed material (LT ); k s = equivalent roughness of Nikuradse (L); I d = index of agreement;
RMSRE = root mean square relative error; and
R = discrepancy ratio. -14-