HYPERCONCENTRATED FLOW AND SEDIMENT TRANSPORT AT STEEP SLOPES

By Dieter Rickenmann1

ABSTRACT: In order to simulate a fine-material slurry of a debris flow, a clay suspension of various concentrations was recirculated in a steep flume. The effect of an increasing fluid density and viscosity on the flow behavior and the bed-load transport capacity of the flow was examined. Viscous effects were found to become important below a limiting particle Reynolds number of about 10. Above this limiting value, density effects cause an increase in the bed-load transport rates as compared to similar conditions with clear water as transporting fluid. The exper­imental data in this range can be described with conventional (Newtonian) formulas and is analyzed together with other bed-load transport data. Two different calcu­lation schemes are proposed for the steep slope range (S >~ 10%) where the bed-load concentration is significant with respect to the total flow depth. Below the critical particle Reynolds number of 10, the bed-load transport rates decreased strongly.

INTRODUCTION

In a torrent catchment, different sediment transport processes may occur. During a flood event, increasing discharges can destroy the armor layer of the torrent bed and fluvial transport of bed material will start. In addition there may be sediment supply to the channel from slope failures (slides and undercutting of banks) and thus, sediment availability could be sufficient for the flow to move sediment in rates close to its transport capacity. In steep reaches of the channel, saturated parts of loose material may be set in motion rather suddenly and a debris flow can form. Then the flow of this sediment-water mixture is highly unsteady; debris flows usually consist of one or several pulses or waves. At the front of the flow, the particles are more or less uniformly distributed over the flow depth, while the mixture behind the front becomes more and more diluted, and at the end, the coarser particles are concentrated near the bed as in fluvial sediment transport (e.g., Costa 1984; Davies 1988).

The water and the suspended fine sediment may be considered as a new fluid matrix that can have a different rheological behavior than that of a Newtonian fluid like clear water (Johnson 1970; Costa 1984). In the tran­sition region between normal streamflow (with fluvial sediment transport) and a debris flow, increasing amounts of fine material form a hypercon­centrated slurry. These conditions may also be present in the rear, more diluted part of a debris-flow wave.

Here a hyperconcentrated flow is defined as a still more or less uniform flow with a high suspended fine-material concentration and a viscosity larger than that of water; coarse sediment may be transported as bed load. The term debris flow is restricted here for an unsteady, pulsing flow of a mixture of water and both coarse and fine sediment. To distinguish between these

•Hydr. Engr., Amt fur Gewasserschutz und Wasserbau des Kantons Zurich, Walchetor, 8090 Zurich, Switzerland.

Note. Discussion open until April 1, 1992. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 13, 1990. This paper is part of the Journal of Hydraulic Engineering, Vol. 117, No. 11, November, 1991. ©ASCE, ISSN 0733-9429/91/0011-1419/$1.00 + $.15 per page. Paper No. 26320.

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two flow types, a sediment concentration by volume between about 30% and 60% was proposed by various researchers; a comparison of different classification schemes including such limiting values is given by Bradley (1986).

In the past, studies on sediment transport in rivers and flumes were mainly concerned with bed slopes up to a few percent. It is only recently that interest on sediment-transport processes in steep channels has grown. Such studies, including flume slopes in excess of 10%, were performed by Mi-zuyama (1977), Mizuyama and Shimohigashi (1985), and Smart and Jaeggi (1983). The transporting fluid used in these flume tests was clear water. In torrents where debris flows also occur, the influence of large amounts of suspended fine particles may become important. Accordingly, the objective of the present study was to examine the effect of an increasing density and viscosity of the fluid on the flow and the sediment-transport behavior. A series of steep flume tests was performed using a clay suspension to represent the fine slurry of a debris flow, and the flow resistance and the sediment-transport capacity were determined for various clay concentrations. The experiments of Smart and Jaeggi (1983) [see also Smart (1984)], which were carried out at the same hydraulic laboratory, served as a reference condition for the clear-water case.

EFFECT OF HIGH FINE MATERIAL CONCENTRATIONS

In the past, two principally different debris-flow theories have been de­veloped. One of them is the viscoplastic model that was applied to flows containing a lot of fine material in a viscous slurry. It is based on the approach by Bingham (1922) to describe a particular type of non-Newtonian flow. The Bingham model also applies to clay suspension flows as used in this study and it can be written as:

T = T* + T1*(!) ( 1 )

where T = the shear stress in the fluid; iB = the (Bingham) yield stress; r\B = the Bingham viscosity; and dvldy = the velocity gradient perpendic­ular to the flow direction.

Considering mainly the interaction between the coarser particles in a debris flow, Takahashi (1978) proposed an alternative model based on Bag-nold's (1954) concept of dispersive pressure. Combining these two basic approaches, more complex models were formulated, for example, Chen (1988a, 1988b) and Julien and Lan (1991). A detailed discussion on debris-flow models can be found elsewhere (Bradley 1986; Rickenmann 1990).

Many hyperconcentrated slurries of fine particles suspended in water behave like a Bingham fluid (Costa 1984; Chen 1986). The flow resistance of these fluids can be analyzed either by a set of equations in terms of the Bingham parameters (Johnson 1970; Naik 1983) or by defining an effective viscosity (xe2 and using conventional Newtonian formulas (Quian et al. 1980; Wan 1982):

(xe2 = T)B + -^ (2)

where h = the flow depth; and V = the mean fluid velocity. Several studies were concerned with the effect of hyperconcentrated flows

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on sediment transport, but they are mainly confined to flatter slopes. There seems to be general agreement that in turbulent hyperconcentrated flows, the suspended load transport rates increase with increasing fine-material concentration; this was concluded both from field observations (Beverage and Culbertson 1964; Bradley 1986) and from laboratory experiments (up­per-regime tests of Simons et al. 1963; Kikkawa and Fukuoka 1969; Wan 1982; Bradley 1986; Wan and Song 1987). With regard to the effect on bed-load transport rates qB, an increase in qB due to an increase of the fluid density is predicted from experimental results by Shields (1936), by Meyer-Peter and Mueller (1948), and by Luque and van Beek (1976), and from theoretical considerations by Yalin (1977) and by Woo (1985). However, Wan and Song's (1987) theory predicts a decrease in the dimensionless bed-load transport rates Q>B for an increasing fluid density at higher shear stresses. Low (1989) found that many bed-load transport formulas adequately ac­count for a change in the solid/fluid density ratio, but that the Smart and Jaeggi (1983) equation should be modified. Information on the effect of an increasing viscosity alone on qB in open-channel flow is less conclusive (Hong et al. 1984; Woo 1985; Lau 1987). Wan (1982) concluded from his experi­ments that <$>B was smaller in the bentonite suspension flows than in the corresponding clear-water flows, particularly in the low flow-intensity region.

There is not yet much information available on sediment transport in laminar flow. In their experiments, both Wan and Song (1987) and Bradley (1986) observed an increase in the total bed-material discharge; in Bradley's study the transport rates in laminar flow were about an order of magnitude higher than in the corresponding turbulent flow (with the same fluid dis­charge). Pipe flow tests reported by Thomas (1979b) may suggest that the bed-load transport rates decrease in laminar flow, assuming that bed load is the predominant mode of transport at a velocity slightly above depositional conditions. Negative lift forces for particles near the bed (Coleman 1967; Davies and Samad 1978) and theoretical considerations by Bagnold (1956) also seem to indicate that bed-load transport should be smaller in laminar flow than in turbulent flow.

EXPERIMENTS

In the experiments, a clay suspension of various concentrations was re­circulated in a 20.1-cm-wide and 5-m-long flume. Commercially available Opalinus clay was added to the water system so as to obtain different clay concentration levels. After settling of the larger components, the concen­tration of the flowing clay suspension could be kept approximately constant. At a given concentration level, a series of tests was performed with different slopes and flow rates. The clay suspension showed increasingly non-New­tonian characteristics with increasing concentration. The maximum density of the suspension was about 1.36 g/cm3 (corresponding to a volume con­centration of 22%), and the maximum effective viscosity |xe2 (defined as analogous to the Newtonian viscosity) reached 1,800 cps.

At each clay concentration level, two different kinds of experiments were performed. In the first series of tests (case A), a fixed rough bed made of gravel was installed to measure the flow resistance of the clay suspension flows without any bed-load transport. In the second series (case B) a mov­able bed of gravel was used, and the same particles were also fed at the flume entrance in order to determine the equilibrium bed-load transport

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rate for a preset slope and flow rate. A rather uniform gravel mixture with a mean diameter d,„ = 1 cm was used as bed material; for this material, d9i) = f .2 cm where dgn is the grain size at which 90% of the material by weight are finer. In the case B tests the bed slope S was varied between 7% and 20%, and the fluid flow rate Q between 10 and 30 1/s. Experiments were performed at five different clay concentration levels; the corresponding fluid properties are summarized in Table 1. The measured flow parameters are given in Table 2 for the experiments without sediment transport (case A). The fixed rough bed in these experiments had the same grain-size characteristics as the bed material used in the sediment. The data of clay suspension flows with bed-load transport (case B) is shown in Table 3. The experimental results labeled HzO were taken from Smart and Jaeggi (1983). (See section on bed-load transport for distinction between case Bl and case B2 experiments.

The Bingham parameters iH and r\B were measured with a rotational viscometer, and the effective viscosity |x(>2 was determined according to (2). The salt tracer technique was used to measure the mean fluid velocity; at the highest concentrations this method was not reliable because of insuffi­cient mixing of the salt solution in the clay suspension (see also remarks in Tables 2 and 3). The flow depth was inferred from a flow surface level determined by ultrasonic distance measuring devices and information about the bed level. In the case B tests, the boundary between nonmoving grains of the bed and the transport layer was detected by electric conductivity readings at different levels across the flume width, since no visual obser­vation was possible in the muddy suspension. All measurement methods are described in more detail in Rickenmann (1990).

In the analysis of both the case A and the case B experiments, a correction for sidewall friction effects was applied. A procedure used by Smart and Jaeggi (1983) was adapted; the method is based on the approach by Einstein (1934) who postulated equal velocities in flow sections with different rough­ness. Thus the reduced flow depth, hr, and the reduced flow rate per unit width, qn were determined from the measured flow depth, H, and the

TABLE 1. Clay Concentration Levels, Ci, and Fluid Properties

a (D

P (g/cm3) (2)

Cr (%) (3)

T« (N/m2) (4)

f\B (cps) (5)

fx,2 (cps) (6)

(a) Case A Experiments (No Bed-Load Transport)

H , 0 CI C2 C3 C4 C5

0.998 1.078 1.165 1.238 1.324 1.365

0.0 4.7

10.0 14.4 19.6 22.1

0.0 0.44 2.82 7.29

23.6 40.8

1.02 1.93 3.60 5.92

20.0 34.3

1 5-10

25-55 60-140

180-400 250-1800

(b) Case B Experiments (with Bed-Load Transport)

H , 0 CI C2 C3 C4 C5

0.998 1.072-1.096 1.141-1.160 1.201-1.246 1.257-1.293 1.356-1.363

0.0 4.4-5.8 8.6-9.7

12.2-14.9 15.6-17.8 21.6-22.0

0.0 0.43-0.67 2.20-2.77 4.31-8.55 12.8-20.1 33.6-40.8

1.02 1.87-2.17 3.00-3.33 4.35-6.74 11.2-16.2 28.8-34.3

1 10-15 35-65

100-200 200-450 800-1000

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measured flow rate per unit width, g, as follows (Rickenmann 1990): hr = H - 2 • [V/(kh\/S)Y5 • (HIB) and qr = q - V • (H - ft,.), where /c,„ is the Striekler value characterizing the flume wall (Ar,v = 110 m1/3/s was used for the perspex side walls) and B denotes the flume width.

FLOW RESISTANCE

Flow resistance measurements were performed for both cases A and B experiments. The case A test series allowed an analysis of the clay suspen­sion flows alone, without any interference of a two-phase flow as in experi­ments B.

The flow resistance data was plotted on a Moody-type diagram in terms of the Darcy-Weisbach friction factor/ = 8T„/(pV2) and the Reynolds num­ber R2 = 4V77,.p/m2, defined with the effective viscosity (jie2. Here T0 = pgh,S = the bed shear stress; p = the fluid density; g = the gravitational acceleration; and S = the bed slope. Some of the clay suspension flows without bed-load transport (case A) were in the laminar region, and the corresponding data points lie close to the relation for laminar Newtonian flow, / = 96/R2 [Fig. 1(a)]. This confirmed the findings of other studies (Quian et al. 1980; Wan 1982) that by using the effective viscosity \x.e2, the flow resistance may be described by conventional Newtonian formulas.

The data was also analyzed by using a Colebrook-type flow resistance equation:

$!•*»{$+* <3> where ks = the equivalent sand roughness; (3 = a coefficient depending to some extent on the relative flow depth; and A is constant in the hydraulically fully turbulent region where no viscous effects are present (Yalin 1977). In Fig. 2, the data is plotted in terms of A' = [1/V/ — 2 • \og(hrld9Q)] and RJ, and R2 = v*dp/\ie2 is the particle Reynolds number, with v* = (gh,.S)0-5

and d = dgo. It is seen that viscous effects only become important for R2 :£ 10. Above this critical value, A' is approximately constant with an average of 2.4; from a comparison with (3) it can be concluded that a value of ks = 0.76 • d90 would best describe the clay suspension experiments in the turbulent region.

In the experiments with bed-load transport (case B), no flows in the laminar region were observed. The analysis of the flow resistance data shows only a very slight increase of the friction factor / for some of the C2 and C3 tests [Fig. 1(b)]. This minor change is associated with an increase in bed-load concentration and thus, flow depth, and is not a direct effect of altered fluid properties.

In further analysis, a search was made for a prediction equation for the fluid velocity that would apply to both the bed-load transport experiments in the fully turbulent clay suspension flows (for which R* > 10, case Bl, see also section on bed-load transport) and the bed-load transport tests of Smart and Jaeggi (1983).

Jaeggi (1983) proposed a modified flow resistance equation for a sediment transporting flow over a movable bed at smaller relative depths hldgo, be­cause many conventional formulas tend to overpredict the velocities in the range 5 < h/d9n < 20. His equation is given as:

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TABLE 2. Case A Experiments, No Bed-Load Transport

a 0)

H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H , 0 CI CI CI CI CI CI CI CI CI CI CI CI C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C3 C3 C3 C3 C3 C3 C3 C3 C3 C3 C3 C3

S (2)

0.05 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20 0.05 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20 0.05 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20 0.05 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20

Q (l/s) (3)

10.0 20.0 40.0 10.0 20.0 30.0 10.0 15.0 30.0 10.0 20.0 40.0 10.0 20.0 40.0 10.0 20.0 30.0 10.0 15.0 30.0 10.0 20.0 40.0 10.0 20.0 40.0 10.0 20.0 30.0 10.0 15.0 30.0 10.0 20.0 40.0 10.0 20.0 40.0 10.0 20.0 30.0 10.0 15.0 30.0 10.0 20.0 40.0

V (m) (4)

1.20 1.62 2.13 1.46 1.95 2.27 1.63 1.95 2.63 1.76 2.35 3.18 1.18 1.69 2.35 1.42 1.98 2.41 1.60 1.90 2.65 1.77 2.39 3.36 1.15 1.64 2.18 1.19

• 1.97 2.42 1.54 1.87 2.61 1.69 2.38 3.24 1.22" 1.66a

2.23a

1.43 1.90 2.30 1.58 1.85 2.51 1.73 2.28 3.16

H (cm) (5)

4.01 6.02 9.38 3.19 5.04 6.44 2.83 3.63 5.83 2.58 4.24 6.29 3.98 5.86 8.45 3.08 4.79 6.03 2.72 3.44 5.38 2.44 3.78 5.90 3.99 5.87 9.10 3.09 4.76 6.02 2.70 3.42 5.36 2.36 3.68 5.82 3.59 5.90 8.89 3.09 4.84 6.00 2.89 3.66 5.56 2.53 3.79 5.68

Tfl

(N/m2) (6)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 2.92 2.92 2.92 2.51 2.51 2.51 2.92 2.92 2.92 2.92 2.92 2.92 8.65 8.65 8.65 6.60 6.60 6.60 6.95 6.95 6.95 6.95 6.95 6.95

•<\B

(ops) (7)

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 3.62 3.62 3.62 3.55 3.55 3.55 3.62 3.62 3.62 3.62 3.62 3.62 6.70 6.70 6.70 5.46 5.46 5.46 5.75 5.75 5.75 5.75 5.75

' 5.75

P (g/cm3

(8)

0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 1.079 1.079 1.079 1.079 1.079 1.079 1.079 1.079 1.079 1.079 1.079 1.079 1.165 1.165 1.165 1.163 1.163 1.163 1.165 1.165 1.165 1.165 1.165 1.165 1.250 1.250 1.250 1.227 1.227 1.227 1.237 1.237 1.237 1.237 1.237 1.237

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TABLE 2. Continued

a (D

C4 C4 C4 C4 C4 C4 C4 C4 C4 C4 C4 C4 C5 C5 C5 C5 C5 C5 C5 C5 C5 C5 C5 C5

S (2)

0.05 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20 0.05 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20

Q (l/s) (3)

10.0 20.0 40.0 10.0 15.0 30.0 10.0 15.0 30.0 10.0 20.0 40.0 10.0 15.0 20.0 10.0 15.0 20.0 10.0 15.0 20.0 10.0 20.0 30.0

V (m) (4)

1.18° 1.78" 2.41" 1.50" 1.87" 2.46" 1.73" 2.00" 2.66" 1.82" 2.40" 3.36" 0.75" 1.14" 1.46" 1.13" 1.54" 1.86" 1.42" 1.77" 2.07" 1.74" 2.27" 2.76"

H (cm) (5)

4.20 5.58 8.27 3.31 4.00 6.06 2.87 3.73 5.62 2.74 4.14 5.92 6.69 6.53 6.83 4.43 4.85 5.37 3.50 4.21 4.80 2.87 4.39 5.41

Tfl

(N/m2) (6)

23.60 23.60 23.60 23.60 23.60 23.60 23.60 23.60 23.60 23.60 23.60 23.60 40.80 40.80 40.80 40.80 40.80 40.80 40.80 40.80 40.80 30.00 30.00 30.00

1)B

(cps) (7)

20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 34.30 34.30 34.30 34.30 34.30 34.30 34.30 34.30 34.30 26.00 26.00 26.00

P (g/cm3)

(8)

1.324 1.324 1.324 1.324 1.324 1.324 1.324 1.324 1.324 1.324 1.324 1.324 1.365 1.365 1.365 1.365 1.365 1.365 1.365 1.365 1.365 1.340 1.340 1.340

"In these tests, velocity measurement by salt tracer technique was not reliable (C3, C4), or not possible at all (C5); the velocity determined as V = Q [B • H) was used here.

7 =2" L1 - e x p U^V where the coefficients (^ and ax depend on the grain-size distribution and the packing and the shape of the bed material. For their steep channel bed-load transport tests, Smart and Jaeggi (1983) determined o^ = 0.05 and Bt = 15. Using the same coefficients, the performance of (4) is shown in Fig. 3 for the steep flume data sets of Smart and Jaeggi and of the writer. It is observed that (4) tends to slightly overpredict the velocities of the clay suspension tests. It may be noted that (4) was developed from experimental data with flatter slopes; for large relative flow depths and small slopes, the correction factor in square brackets of (4) approaches unity and the formula becomes similar to a Nikuradse-type equation.

Based on the concept of dispersive stresses between the moving grains, Takahashi (1978) proposed an equation for the mean velocity of a quasi-steadily moving debris-flow front. Replacing concentration and density terms together with grain-shearing coefficients by one single parameter A*, the following expression is obtained (Takahashi 1978):

In 12.27ft

M9o (4)

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TABLE 3. Case B Experiments with Bed-Load Transport

a (D

s (2)

Q (l/s) (3)

V (m/s) (4)

H (cm) (5)

O f l

(kg/s) (6)

f «

(N/m2) (7)

T}B

(cps) (8)

P (g/cm3)

(9)

(a) Case Bl Experiments

H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H 2 0 H , 0 CI CI CI CI CI CI CI CI CI CI CI CI C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C3 C3 C3 C3 C3 C3 C3 C3 C3 C3

0.07 0.07 0.10 0.10 0.10 0.15 0.15 0.15 ' 0.20 0.20 0.20 0.20 0.07 0.07 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20 0.20 0.07 0.07 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20 0.20 0.20 0.07 0.07 0.10 0.10 0.10 0.15 0.15 0.15 0.20 0.20

15.0 " 25.0 10.0 15.0 30.0 10.0 15.0 30.0 10.0 15.0 20.0 30.0 15.0 25.0 10.0 15.0 30.0 10.0 15.0 30.0 10.0 15.0 20.0 30.0 15.0 25.0 10.0 15.0 30.0 10.0 15.0 30.0 10.0 15.0 20,0 30.0 15.0 25.0 10.0 15.0 30.0 10.0 15.0 30.0 10.0 15.0

1.28 1.55 1.07 1.38 2.06 1.18 1.57 2.52 1.26 1.74 2.44 2.59" 1.16 1.57 1.11 1.35 2.01 1.18 1.52 2.03 1.25 1.60 1.90 2.47 1.27 1.46 1.12 1.40 2.18 1.18 1.58 2.22 1.29 1.83 2.19 2.75 1.21 1.60 1.15 1.41 1.93 1.23 1.61 2.34 1.58 1.73

5.90 8.20 5.00 6.30 6.00 4.80 6.00 8.00 6.00 6.80 7.60 8.50 5.19 7.31 4.61 5.25 7.69 4.82 6.01 8.12 5.62 6.89 7.77 9.07 5.83 7.80 4.80 5.96 8.54 4.95 6.30 8.52 6.58 7.70 8.25 9.79 5.62 8.11 4.50 5.91 8.72 5.44 6.70 9.04 7.46 8.69

0.68 1.42 0.84 1.60 3.53 2.30 4.08 9.49 4.80 8.84

10.30 14.91 0.88 1.49 1.49 2.31 4.34 3.52 5.29 9.08 7.05

10.58 12.88 17.63 1.15 1.97 2.03 3.12 5.08 4.75 7.05

10.85 9.63

14.37 16.27 20.74

1.49 2.24 2.58 3.91 4.95 5.97 9.01

12.20 13.56 17.55

0.00 0.00 0.00 0.00 0.00 0,00 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.61 0.43 0.60 0.63 0.59 0.52 0.63 0.59 0.58 0.43 0.63 2.70 2.56 2.50 2.77 2.26 2.20 2.20 2.41 2.50 2.38 2.26 2.26 8.06 4.31 8.07 8.07 7.35 6.95 8.14 7.75 8.55 8.14

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 2.17 2.09 1.93 2.13 2.13 2.05 1.93 2.13 2.17 2.03 1.93 2.13 3.33 3.22 3.32 3.33 3.06 3.16 3.16 3.19 3.32 3.22 3.00 3.06 6.27 4.35 6.43 6.43 6.00 5.75 6.74 6.37 6.10 6.74

0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 1.092 1.084 1.075 1.084 1.087 1.091 1.072 1.088 1.086 1.096 1.077 1.088 1.152 1.149 1.160 1.153 1.145 1.153 1.152 1.147 1.158 1.159 1.141 1.145 1.245 1.201 1.243 1.243 1.223 1.240 1.241 1.228 1.246 1.236

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TABLE 3. Continued

a (1)

.V (2)

Q (l/s) (3)

V (m/s) (4)

H (cm) (5)

GB

(kg/s) (6)

T/J

(N/m2) (7)

Tie (cps) (8)

P (g/cm3)

0) (a) Case Bl Experiments-Continued

C4 C4 C4 C4

0.07 0.10 0.15 0.15

25.0 30.0 10.0 30.0

1.69 2.08 1.32 2.49

7.06 8.15 6.21 8.53

2.39 7.18 6.38

13.32

18.60 17.60 20.10 16.20

16.20 15.70 17.50 14.80

1.286 1.275 1.266 1.257

(b) Case B2 Experiments

C4 C4 C4 C5 C5 C5 C5 C5

0.07 0.10 0.10 0.07 0.10 0.10 0.15 0.20

15.0 10.0 15.0 15.0 10.0 15.0 10.0 10.0

1.36 1.16 1.43 1.36b

1.16" 1.43b

1.32b

1.58b

5.83 4.90 6.00 6.85 5.75 6.50 6.60 7.80

1.20 1.76 3.35 0.59 0.81 2.17 3.06 7.65

18.60 20.10 16.20 40.80 40.80 40.80 33.60 33.60

16.20 17.50 14.80 34.30 34.30 34.30 28.80 28.80

1.287 1.293 1.275 1.363 1.363 1.363 1.356 1.356

"No velocity was measured in this experiment; velocity calculated by Smart and Jaeggi (1983) was used here.

bIn these tests, velocity measurement by salt tracer technique was not reliable (C5); as approximation, velocity of corresponding flow at next lower concentration level (C4 or C3) was used here.

U

(gdf

3L gd3

^5 = A*(sin a)"2 (5)

where U = the mean velocity of the moving grains; q = the fluid discharge per unit flume with; and a = the bed slope angle. Takahashi plotted his experimental results in terms of the nondimensional velocity [given by the left-hand side of (5)] versus the bed slope tan a. As predicted by (5), there is only a slight dependence on tan a in his figure. From his diagram (Tak­ahashi 1978), it can be determined that A* ~ 1.3. Using this experimental value for the parameter A*, (5) can be transformed into:

U = 1.3 (sin af-2q°-6g0-2

d0A (6)

Interestingly a regression analysis with the two steep flume data sets dis­cussed previously results in an empirical relation very similar to (6). In fact, Fig. 4 shows good agreement for the steep flume data between measured and calculated fluid velocities; putting sin a ~ tan a = S, d = d9Q and U = V, (6) becomes:

'0.2„0.6n0.2

V 13S^q°-6g

d0' «Q<)

(7)

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0.40 0.30

0.20

f

0.10 0.08 0.06

0.04 0.03

I c, [% T

22.1

\ ^ \

. \

\ \

11 96

f " Re

i \

. l ^ * - - ^ ^ * s X, ^ \ \ N , \

\ \ V N

\

* 19.6

\ I

-tr~

\ •>.

% V N

• »

+

14.4

0

10.0 *

4.7 *

0.0

f S = 20°/c 1 J-

]S=15%f.

fS = 5%]

<\ »\ X

\ \

\

1-

"

~\ ->!

^ i *

*

Sl~A

\ ~ «< ̂ \r

102 103 10" 105

Re2 10s

C, [%]

0.80 0.60

0.40 0.30

0.20

f

0.10 0.08

\ \ / ' \

V \

-r

\ A

4 .

\

X

1

V

96

\ ^

s? \

^ \ \ + - N

*

x

\

^

\

\

^

\ \

\

1

| S = I5%[ | | |

\ |s-m%l\ \L-\ V

^

V

\

\

\

| S = 5°/

n • L n "—

kv xa ^ \ •u

5^20%]

I ^r-

\

ft • *

V

102 103 104 105 „ 106

Re2

FIG. 1. Flow Resistance Data of (a) Clay Suspension Flows over Fixed Rough Bed; and (b) Experiments with Bed Load Transport. Data Points with Equal Slope are Connected by Straight Lines

The correlation coefficient squared is r2 = 0.79 and the standard error is SE = 14% of the mean measured velocity. It is remarkable that the fluid velocity of a bed-load transporting flow at slopes 0.05 < = S < = 0.20 may be predicted by the same formula as the velocity of the front of a debris flow. The corresponding debris-flow experiments of Takahashi (1978) cover a slope range 0.17 < = S < = 0.47.

BED-LOAD TRANSPORT

In the case of bed-load transport in the clay suspension (case B), each experiment was performed with the same preset slope and flow rate as for the clear-water tests by Smart and Jaeggi (1983), which served as a reference condition. The ratio of the corresponding bed-load transport rates per unit width, qB/qB,Hicn w a s found to increase with increasing clay concentration of the suspension (and thus, fluid density) up to a limiting value, above which transport rates decreased strongly. This change in bed-load transport capacity can also be plotted against the particle Reynolds number R* (Fig. 5), which represents the effect of a changing fluid viscosity; here RJ is determined with the mean grain size dm. It can be concluded from Fig. 5

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1 hr A ' = T p -2 log d 9 0

«

ft

r T

Y

r

_|~Re2*

rl T

I C, [%]| 22.1

"++ «

Y

t-

+ i+

i 0

1 1 1

1 1

= 10[]Re2* = 5 1 1

* 19.6

[»

a •

+

14.4 10.0

* * * ***

^ 4.7

A A < £ . A

0.0

0 1 2 3 4 log (Re2*)

FIG. 2. Flow Resistance Data of Clay Suspension Flows over Fixed Rough Bed, Shown in Terms of Parameters A' = [l/(Vf - 2 • log(hJdmi)] and K

that the strong decrease in transport rates occurs for values of R* below about 10-15.

For open-channel flow, the thickness of the laminar (viscous) sublayer, k, is given as (Yalin 1977): X = 11.6Wu* = 11.6(T|/p)/w*, where v is the kinematic and T| is the dynamic (Newtonian) viscosity. This implies that in a flow with R* ~ 12, the laminar sublayer is of the same magnitude as the equivalent sand roughness ks. It is assumed that the relationship for X is also valid in the case of the clay suspension, using the effective viscosity |xe2. With regard to the critical value of R* shown in Fig. 5, this implies that the thickness of the laminar sublayer is of the order of the grain size of the transported bed material. Thus it appears that once the flow around the grains becomes laminar, the bed-load transport capacity starts to de­crease, for otherwise equal flow conditions.

Interestingly, also for the clay suspension flows over fixed rough bed (case A), a limiting value of RJ = 10 was found below which viscous effects become important. It is therefore concluded that in the bed-load transport experiments, the decrease in qB for RJ <~ 10 (case B2) is due to the increasing viscosity while viscous effects are negligible in the region R* > ~ 10 (case Bl).

Transport Equation, Steep Slopes In the following analysis, only those clay-suspension bed-load transport

experiments with negligible viscous effects (case Bl) are considered. It was found that the transport rate qB can be described by a simple relationship including the (corrected) fluid flow rate qr, the bed slope S, and the param­eter (s — 1), where s = cr/p is the ratio of the grain density to the fluid density.

A bed-load transport equation of a similar form was already proposed by Smart and Jaeggi (1983) from their steep flume tests (slope range 3% < S < 20%) in which clear water was used as transporting fluid:

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Vcoic.[m/sl

Vmeas.[m/S]

FIG. 3. Comparison between Measured Fluid Velocities, V„„m, and those Calcu­lated with (4), V„k, for Two Steep Flume Data Sets

- ^ fePM-fc ) ; <» where d30 = the grain size at which 30% by weight of the material is finer; 6„ = the critical Shields stress at initiation of motion; and 9„, = hrjn • SI [{s — 1) • dm] — the dimensionless bed shear stress determined with the mixture flow depth hrm (including the space occupied by the transported solids but corrected for sidewall friction) and the mean grain size d,„. It is noted that (8) was developed in order to predict also the bed-load transport rates of the Meyer-Peter and Mueller (1948) experiments covering a slope range of 0.04% < S < 2% [data given by Smart and Jaeggi (1983)].

The data of the clay-suspension experiments (case Bl) was analyzed to­gether with the Smart and Jaeggi tests, and a regression calculation yielded the following transport equation, valid for the slope range 5% < X < 20%:

' • - < r r ^ ( £ ) " < ' ' " ' - ^ (9)

where qcr = the critical discharge at initiation of motion. The correlation coefficient squared is r2 = 0.95 and the standard error is SE = 20% of the

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Vcalc.[m/S]

2.5

2.0

1.5

1.0

0.5

£

J

A ^

*X

A &

A /

A / X

Oy/ x

* "

Data set

Smart/Jaggi

Rickenmann

X A

A

X . /

4 " / *

A * / * ft

«

Uniform grain size

A

*

Mixture

x

I

-

0.5 1.0 1.5 2.0 2.5 3.0 Vmeas.[m/S]

FIG. 4. Comparison between Measured Fluid Velocities, Vmem, and those Calcu­lated with (7), V,.„,c, for Steep Flume Data

qB

3.2

2.8

2.4

2.0

1.6

1.2

0.8

qB,H20

T

"f >

s

1 ft

"-••-J

V-':"

n J

-i

5

+

+

0

<5

"

<

0

*

T

ft

+

« *

* >

* * * + /

*

1 1 c, [%]

21.6-22.0 15.6-17.8 12.2-14.9 8.6-9.7 4.4-5.8

1.0 10.0 100.0 Re2*

1000.0

FIG. 5. Ratio of Bed Load Transport Rate Measured in Clay Suspension to Value Obtained in Clear Water, qBlqBM20, as Function of Grain Reynolds Number R*

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1

mean measured transport rate. The critical discharge qcr is expressed by a relationship proposed by Bathurst et al. (1987), which was slightly modified to include also the density factor (s - 1) (Rickenmann 1990):

q„ = 0.065(5 - l)1-67g°-srfi6S'S-1-12 (10) where d50 = the grain size at which 50% by weight of the material is finer.

By comparing (8) and (9), some differences can be observed. First, the exponent of the density factor (s - 1) is larger in the second formula. This is not surprising since no variation of the density ratio was included in the Smart and Jaeggi (1983) tests. Apart from the clay-suspension data of this study there is other evidence that the exponent of (s - 1) should be larger than in (8). Low (1989) conducted a large number of experiments with lightweight sediments to investigate the effect of a change in the density ratio s on bed-load transport rates. He used water as transporting fluid, and thus, no possible viscous effects were involved as in this study. He examined his results in terms of the parameters used by Smart and Jaeggi (1983) and concluded that the exponent of the density factor (s - 1) in (8) should be increased to 1.5 (and the constant adjusted) to obtain best agreement with his data. An improved version of the Smart/Jaeggi equation (Rickenmann 1990) also better predicts some data of Meyer-Peter and Mueller (1948) which included transport tests with coal (s = 1.25) and barite (s = 4.22) as bed load.

A second difference concerns the influence of the slope parameter. In the analysis of this study, it was found that the exponent of S should be larger for the steep flume tests than for the flatter slope conditions of the Meyer-Peter and Mueller experiments. In the development of (8), the num­ber of tests with a slope S s 5% was considerably larger than the number of tests for the steeper slope range. This may explain why the exponent 1.6 in (8) is close to the exponent of 1.5, which results from a regression analysis of the Meyer-Peter and Mueller experiments alone (Rickenmann 1990). It is interesting that both Mizuyama (1981) and Mizuyama and Shimohigashi (1985) also proposed a dependency of qB on S2, based on steep flume experiments in a similar slope range as in this study.

To describe the critical conditions at initiation of motion, the use of (10) has the advantage that no information of the flow depth is needed to de­termine the critical discharge qcr. The bed-load transport rate is given as a function of the flow rate (q) in both (9) and (8). With (9), only the slope, bed material, and fluid characteristics are needed as further input parameters whereas (8) requires in addition the use of a flow resistance relationship in order to determine 0m.

Transport Equation, Steep and Flat Slopes As an alternative form to (8) or (9), an analysis was made of both the

steep flume experiments (clay-suspension data of the author and clear-water tests of Smart and Jaeggi) and the low slope data of Meyer-Peter and Mueller in terms of the dimensionless parameters $>B and 8,„, where the dimension-less bed-load transport rate is defined as $ B = qB/[g(s - l)^,]1-2 . In this study, the following equation was determined by regression analysis to give best agreement between measured and predicted transport rates for the entire slope range of 0.1% <= S <= 20%:

*° = j^{tY^-«')F1A ' (11> 1432

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where F = V7(g • /?,..,„) = the Froude number, determined with the mixture flow depth /(,.,„. The critical Shields stress at initiation of motion was taken as 0.047 for the Meyer-Peter and Mueller experiments, while for the steep flume tests this value was somewhat reduced by a slope correction according to Stevens et al. (1976). With (11), the correlation coefficient squared is r2

= 0.96 and the standard error SE = 35%, for a total number of 252 ex­periments. It is interesting that (11) predicts both the steep and low slope data almost as well as do separate formulas valid only over a limited slope range. Considering the steep flume data only (115 experiments), as used in the development of (9), the correlation parameters for (11) are: r2 — 0.96 and SE = 25%. The performance of (11) over the entire slope range can also be judged from Fig. 6, where calculated transport rates are shown against measured values on a log scale. It is seen that the scatter is consid­erably larger in the low slope region (data of Meyer-Peter and Mueller); in this region the flow conditions are much closer to the critical conditions at incipient motion, which makes prediction of transport rates much more difficult.

In comparison, an improved version of the Smart and Jaeggi transport equation [(8)], with the exponent 1.5 instead of 1.0 for the density factor (s — 1), gives a similar correlation (r2 = 0.96) and a slightly greater standard error (SE = 44%) when applied to the entire data set (252 experiments). If the whole slope range could be as well described by a single formula in the form of (8) as in the form of (11), the standard error should rather be smaller since the flow rate could be much more accurately measured in the experiments than the flow depth. However, by the use of hrm in (11) (con-

l°g(qB,caic.[m3/s m])

X

J

' , / I

• s>

y< i

o

s°

!

43 i

& : 1

o

/i*

i *

I'rf f e

Data set

Smart/Jaggi

Rickenmann

M 1

tiller C I r~

4m

Uniform grain size

* *

, «*/

Mixture

*

1 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

l°g(qB,meas.[m3/s m])

FIG. 6. Comparison between Measured Bed Load Transport Rates, <7B,mcas., and those Predicted by (11), qB,mlc_, for Three Data Sets Covering Slope Range 0.1% <= 5 <= 20%

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tained in the parameters 0,„ and F), there is also some information included on the flow resistance. With (8) and (9), on the contrary, the parameter qr = V • hrf contains only the fictitious flow depth hrf, thus not taking into account the increased flow depth due to high bed-load concentrations at steep slopes. These findings are consistent with the aforementioned con­clusion that in the form of (8) and (9) the exponent of the slope factor should be different in the steep and low slope region. To distinguish between the two regions, a limiting slope of about 10% may be assumed above which the space occupied by moving grains is significant with respect to the total flow depth.

DISCUSSION

Increasingly clay concentrations in the clay-suspension experiments of the author produced an interesting result: For a given slope and flow rate, the bed-load transport rates increased considerably with increasing fluid density, particularly at steeper slopes. This change was associated with only a minor increase in flow depth and in flow velocity, implying that the flow resistance did not significantly change [Fig. 1(b)]. It can be concluded that the fluid velocity is largely determined by the slope and the flow rate. Since these two parameters also determine the bed-load transport rates [see (9)], any effects of bed load on the fluid velocity could be implicitly accounted for by q and S. This may explain the rather good performance of (7), in which the fluid velocity is related to the slope, the flow rate, and a characteristic grain size.

As a rough approximation, it is proposed that the turbulent flow in a steep rough channel is mainly governed by the slope, the flow rate, and the fluid density. For a given q and 5, larger depths were observed for the flows over a movable bed than for the flows over a fixed bed, while the fluid velocities were only slightly higher in the latter case. These larger flow depths (or shear velocities) are mainly associated with the space required for the transported grains. The only minor difference in fluid velocity suggests that the adjustment of the (mixture) flow depth could be the primary mechanism by which the flow supplies the energy required for grain movement. It may be hypothesized therefore that the fluid-grain mixture is a self-regulating system, adjusting flow depth and transport rates in such a way that the fluid velocity is not greatly affected, but instead is largely determined by the slope and flow rate.

At clay concentrations Q above about 17% (p > = 1.28 g/cm3), a decrease in the bed-load transport rates was observed with further increasing de­values. This decrease was found to be associated with a grain Reynolds number R2 below about 10 (Fig. 5), which implies that the thickness of the laminar sublayer, \ , is greater than the grain size, d. A similar conclusion can be drawn based on a study about the hydraulic transport of solids in pipes (Thomas 1979a, 1979b) in which the transport capacity can be expected to decrease if \ > ~ d. Negative lift forces may be an explanation for this decrease. Based on experiments, Davies and Samad (1979) found the lift forces to decrease with decreasing particle Reynolds numbers; they meas­ured negative values for R* <~ 5. Similar experimental results were re­ported by Coleman (1967).

It is noted that the case B2 experiments of this study were in the hy-draulically smooth turbulent regime. For a given pressure gradient in pipe flow, Thomas (1979b) concluded that the transport capacity should be smaller in laminar than in turbulent flow. Since the flow resistance increases strongly

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in laminar flow with increasing viscosity, larger flow depths will occur in open channel flows (for a given flow rate), implying an increase in shear stress. With increasing viscosity, the bed-load transport rate is expected to decrease; however, a part of the grains may then be transported in suspen­sion (not supported by turbulence but due to a very small settling velocity). It is difficult to predict what the combined effect will be on the bed-load transport capacity in laminar open-channel flow. Experimental evidence (Bradley 1986; Wan and Song 1987) suggests that the total transport capacity may be considerably higher in laminar flow.

Three mechanisms were proposed by Takahashi (1981) for the initiation of a debris flow: (1) A landslide from a hill slope may transform into a debris flow; (2) at sufficiently steep slopes (>~ 27%) and with large bed shear stresses, material deposited in a channel bed may be mobilized and eventually form a debris flow; and (3) accumulation of debris material from the side slopes can cause a temporary blockage of material to be built up in the channel, and a debris flow may be initiated when this material starts to move.

It was noted by Davies (1988) that a temporary blockage of large grains can occur in a uniform flow of a grain-fluid mixture in a steep, narrow channel. He further pointed out that the development of roll waves may cause the pulsing of a debris flow; the critical Froude number for the oc­currence of roll waves is much smaller in laminar than in turbulent flow. Based on field data, Davies showed that the limiting conditions developed for clear-water flows also apply generally for debris-flow conditions. Re­cently, Savage (1989) demonstrated theoretically that the critical Froude number for the occurrence of roll waves decreases with increasing cohesion (Bingham yield stress), viscosity and particle interaction.

In the light of the experimental results of this study, the pronounced decrease of the bed-load transport rate in macroviscous flow (i.e., having R2 s 10) may also be seen as an element enhancing the instability of a uniform flow, and thus offer a possible explanation for the pulsing behavior of debris flows. In the experiments, macroviscous flow occurred at clay concentrations Cf above about 17%. Scaling considerations suggest that a higher critical value of Cf can be expected in the field (Rickenmann 1990). In the range of fine-material concentrations Cf between 20% and 50%, the Bingham yield stress TB increases strongly with Cf (Chen 1986; O'Brien and Julien 1986). According to field data compiled by Davies (1988), the tran­sition to macroviscous flow may be expected to occur in this concentration range. For a given flow rate in a torrent, a local input of fine material might thus cause a sudden decrease in bed-load transport capacity (at R* — 10), causing a part of the grains to be deposited. If such events are frequent and/ or large enough, the deposited material could form a temporary dam. In contrast to direct deposition of material from the side slopes, a partial or full blockage to the flow might therefore also be caused by a change in flow properties.

CONCLUSIONS

To study the effect of the increasing non-Newtonian properties with in­creasing fine-material concentration (i.e., increasing Bingham yield stress and Bingham viscosity) on the flow, experiments were performed with a clay suspension flowing over a fixed rough bed in a steep channel without any sediment transport (case A). The flow resistance analysis indicated that Newtonian formulas can still be used if an adequate viscosity is defined,

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such as the effective viscosity \xe2. No viscous effects could be detected in the hydraufieally transitional region [10 £ R* < 54, (Fig. 7)]. This is in agreement with the analysis of the bed-load transport tests (Fig. 5). It is interesting that a semitheoretical equation can be applied to both the ex­perimental flows with intense bed-load transport (with slopes up to 20%) and experimental debris flows (with slopes up to 47%).

The case A experiments with R? ^ 10 were in the laminar region, and the corresponding data points lie close to the relation for laminar Newtonian flow [Fig. 1(a)]. It appears that the hydraulically smooth turbulent regime (RJ :S 10) is obscured by the formation of a plug flow in a Bingham fluid. This is a zone with no shearing between adjacent fluid layers, extending from the flow surface downwards; with increasing Bingham yield stress of the fluid, this zone occupies increasing proportions of the flow cross section. When it merges with the viscous sublayer, the flow becomes laminar.

Also bed-load transport experiments were performed with clay-suspen­sion flows in a steep channel (case B). With increasing clay concentration, the bed-load transport capacity increased due to the decreasing solid/fluid ratio s. Once the effect of the increasing viscosity of the fluid becomes important (at R| s 10) and the flow around the grains is laminar, the trans­port capacity decreases markedly with further increasing clay concentration.

The bed-load transport experiments in the turbulent region (RJ < 10) were analyzed together with the data sets of Smart and Jaeggi (1983) and of Meyer-Peter and Mueller (1948). It was found that in the g^-form [such as (9)] the equations for the steep and the lower slope range should have a different exponent in the slope factor. In the $B-form [such as (11)], however, a single equation may be used for the whole slope range since the dimensionless bed shear stress 6,„ also contains some information on the flow resistance increase due to significant bed-load concentrations at steep slopes. As a result of the interaction between bed-load transport and flow resistance in the steep slope range (S >~ 10%), two different calculation procedures are proposed. Since (4) predicts the flow velocity as a function of the flow depth, it may be used in connection with the bed-load transport equation [(11)]. This calculation procedure can be applied over a slope range 0.1% <~ S < ~ 20%. If, however, the flow rate is used as input parameter instead of the flow depth, it is recommended that (7) and (9) are used to calculate the other parameters of the two-phase flow. These formulas should, however, only be applied in the steep flume range, i.e., 5% < ~ S <~ 20%. The two proposed calculation schemes are discussed in more detail by Rick-enmann (1990).

ACKNOWLEDGMENTS

The presented study was performed at the Laboratory of Hydraulics, Hydrology and Glaciology of the ETH in Zurich where the writer worked as a research engineer.

APPENDIX I. REFERENCES

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APPENDIX ! l . NOTATION

The following symbols are used hi this paper:

A = constant in flow resistance equation; A* = parameter in velocity equation for debris-flow front;

B = flume width; Cf = volume concentration of fine material; Ci = clay-concentration level; d = grain size;

d,„ = mean grain size; d30 = characteristic grain size, 30% of the material by weight is finer; d5tt = characteristic grain size, 50% of the material by weight is finer; d90 = characteristic grain size, 90% of the material by weight is finer;

(dv/dy) = fluid velocity gradient (shear rate) perpendicular to flow direc­tion;

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Froude number = (V/(g/hy-2); Darcy-Weisbach friction facioi = (8-iy'pV2); sediment-feeding rate; acceleration due to gravity; measured flow depth; flow depth; (reduced) flow depth corrected for sidewall influence; fictitious fluid flow depth in flow of grain-fluid mixture = (qj V), corrected for sidewall influence; mixture flow depth (including space occupied by moving grains), corrected for sidewall influence; Strickler coefficient for flume walls; equivalent sand roughness; flow rate in flume; volumetric water or fluid discharge per unit width; reduced fluid discharge per unit width, corrected for sidewall influence; critical flow discharge at beginning of bed-load transport; volumetric bed-load transport rate per unit width; Reynolds number defined with effective viscosity |x(,2 = {AVhpl

particle Reynolds number defined with effective viscosity \x.L,2 = (T<*rfp/u,(,,); correlation coefficient (between measured and calculated val­ues); slope (= tan (3); standard error; ratio between grain and fluid density = (cr/p); mean velocity of debris-flow front; average fluid velocity; shear velocity = (ghS)U2\ bed slope angle; coefficient in flow resistance formula of Smart and Jaeggi (1983); coefficient in flow resistance equation; coefficient in flow resistance formula of Smart and Jaeggi (1983); dynamic fluid viscosity; Bingham viscosity; dimensionless bed shear stress = (T„/pg(s — l)d,„) = (hSI(s ~ i )4 , ) ; critical value of 9 at initiation of motion; dimensionless bed shear stress in grain-fluid mixture; = {hrmSI {s - 1)4,); thickness of laminar (viscous) sublayer; effective viscosity = (T|S + iBh/2V); kinematic viscosity; fluid density; grain density; shear stress in fluid; fluid shear stress at bed = (gpSh); Bingham (yield) stress; and dimensionless bed-load transport rate = qB/[g(s ~ l ) ^ ] 1 ' 2 -

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