SIMPLIFIED SHIELDS CONDITION FOR SEDIMENT TRANSPORT INCEPTION

Willi H. Hager¹ and Giuseppe Oliveto²

¹ Professor, VAW, ETH-Zentrum, CH-8092 Zurich, Switzerland

Tel: +41 1 632 4149, Fax: +41 1632 1192, E-mail:hager@vaw.baug.ethz.ch

² Researcher, DIFA, University of Basilicata, I-85100 Potenza, Italy

Tel: +39 0971 205 142, Fax: +39 0971 205 160, E-mail:gndcil21@unibas.it 

Abstract: Based on the Shields diagram and laboratory observations, novel equations for the inception of sediment transport in a nearly straight, almost horizontal rectangular channel are developed. Separate equations for the viscous, the transition and the turbulent rough regimes are provided, that depend exclusively on basic hydraulic parameters such as cross-sectional velocity, that are readily available for the hydraulic engineer. In addition sediment mixtures with an intermediate degree of sediment sizes are considered, and Shields formulation is expanded to account for a mixture densimetric particle Froude number. The results of the present project are readily applicable for design, therefore. 

Keywords: sediment transport, shields diagram, inception of transport, fluvial hydraulics

1    INTRODUCTION

Shields (1936) presented a diagram relating the grain Reynolds number R_*=u_*d_s/n to the so-called Shields parameter T_i=t_i / (r_s-r)gd_s, with u_*=shear velocity, d_s=sediment size, n=kinematic viscosity, t_i=inception shear stress, r_s=sediment density, r=fluid density and g= gravitational acceleration. This relation is presently accepted for the prediction of sediment transport inception, although being subject to critique in the past. Two recent works to clarify Shields’ work were presented by Buffington (1999) and Patel and Ranga Raju (1999), general recent accounts on the problem are by Przedwojski, et al. (1995), and Graf and Altinakar (1996). A historical account on Shields benchmark paper is available (Kennedy 1995).

When applying Shields diagram, the practical engineer faces problems when predicting inception of sediment transport. For a uniform sediment the shear velocity cannot be directly determined whereas the determining size for sediment mixtures has not been thoroughly defined. Often, the shear velocity is eliminated with a uniform flow formula, such as the Manning equation. However, the two additional variables energy line gradient and roughness coefficient then have to be determined. The purpose of the present research was to improve this current drawback mainly relating to practical application, by using basic hydraulic variables to define the inception condition for sediment mixtures. Given that there are many accounts on the Shields condition, no detailed review of present knowledge is given here.

The present research was based on both a theoretical and an experimental approach by which the simplified inception condition is determined. The theoretical framework follows basic hydraulics and results in expressions for the densimetric particle Froude number at inception stage. These results are then verified by controlled hydraulic model tests conducted at VAW, ETH Zurich. The simplified expressions contain only basic variables such as average velocity, grain size and flow depth in a rectangular channel. The effects of gravity and viscosity are accounted for, as also the effect of sediment mixtures, using again the laboratory observations. Examples illustrate the simplified approach.

2    EXPLICIT SHIELDS CONDITION

2.1    General inception equation

The Shields diagram contains the Shields parameter and the grain Reynolds number, in which the first parameter accounts for gravitational, and the second for viscous effects. The shear velocity cannot adequately be determined even in laboratory conditions because probes located close to the sediment surface disturb the bed. Indirect methods involving the velocity profile close to the bed are also questionable for the same reason, except for non-intrusive methods, as also because of the large velocity gradient close to the bed. The free surface slope involves a large degree of uncertainty, given its turbulence pattern that normally allows an accuracy of 1 mm depth reading.

An alternative involves established equations for fluvial hydraulics, namely the uniform flow equation, and a relation accounting for the roughness pattern of a sediment bed. In addition, a simplified Shields diagram is used to inhibit complicated mathematical expressions.

The original data of Shields were not given mathematically but rather a curve was drawn through the data, including a band of confidence. Currently, various equations are available to account for Shields curve, yet most of them are complicated. One of the simpler equations was quoted by Garcia (2000), based on Brownlie. Yet, a simpler approach was recently forwarded by subdividing the domain of interest into three portions, depending on the dimensionless grain size D_*=(g’/n²)^1/3d₅₀ where g’=((r_s-r)/r)g, and n=kinematic viscosity for a water temperature of 25°C, namely (Hager and del Giudice 2000)

T_i = 0.120 D_*^-1/2,    D_* £ 10                          (1)

T_i = 0.026 D_*^1/6,    10 < D_* < 150                     (2)

T_i = 0.060 ,        D_* ³ 150                         (3)

Here T_i=t_50i/(r_s-r)gd₅₀=S_oR_h/r’d₅₀=dimensionless shear stress at inception condition (subscript i), corresponding to the Shields number, and r’=(r_s-r)/r.

For nearly uniform flows, the Manning-Strickler equation can be written as

V = (1/n) S_o^1/2 R_h^2/3                                 (4)

where V=cross-sectional velocity, 1/n=Manning’s roughness coefficient, S_o=free surface slope and R_h=hydraulic radius. According to Strickler 1/n can be expressed with the median grain size as 1/n=21.1d₅₀^-1/6. In dimensionless terms, this is equal to 1/n=6.75 g^1/2d₅₀^-1/6. Using the previous definitions, one can demonstrate that T_i=S_oR_h/r’d₅₀=(nV/R_h^2/3)²(R_h/r’d₅₀), with h=flow depth. Solving for the so-called inception densimetric particle Froude number F_di=V_i/(g’d₅₀)^1/2, Hager and Oliveto (2000) proposed

F_di = 6.75 T_i^1/2 (R_h/d₅₀)^1/6                           (5)

Accordingly, for the three domains of D_* previously specified, one may define a relation between the fluid dynamics (F_d), the sedimentology (T_i), and the relative flow depth (R_h/d₅₀ and thus h/d₅₀). For a given relative flow depth in an almost rectangular channel, one may thus determine the average cross-sectional velocity V_i for sediment transport inception. Such an expression provides basic knowledge for fluvial hydraulics, therefore.

2.2    Simplified inception equations

Inserting (1) for the viscous regime with D_*£10 in (5) gives

F_di = 2.33 D_*^-0.25 (R_h/d₅₀)^1/6                                            (6)

For the transition regime with 10<D_*<150, one has with (2) and (5)

F_di = 1.08 D_*^1/12 (R_h/d₅₀)^1/6                          (7)

For the fully turbulent regime, finally, the result is

F_di = 1.65 (R_h/d₅₀)^1/6                                                 (8)

In the first two cases, the inception densimetric particle Froude number is proportional to a power of the dimensionless grain size D_*, and increases with the relative hydraulic radius, to a power of 1/6. For equal density of sediment and fluid, the inception Froude number may thus be increased when the relative flow depth increases, a fact well known today. Compared to the present process of inception evaluation for sediment-water flows, (6) to (8) are direct and simple relations that may indeed readily be used. In addition, the present formulation involves three main effects, namely those of gravity (F_d), of viscosity (D_*) and of relative flow depth (R_h/d₅₀). The effect of sediment mixtures is discussed based on experimental observations.

3    EXPERIMENTS

3.1    Test facilities

The experiments were conducted in two rectangular channels, of widths B=1.00 m for the main tests, and B=0.5 m to check for the width effect. The main channel had a length of 13 m, with a working section of about 5 m. The bed was always inserted horizontally, and the inception conditions were observed close to the end of the sediment bed, where erosion started. Typical free surface slopes were smaller than 1%, and depended both on flow depth and sediment size. The accuracy using this procedure for evaluating inception conditions was thought to be better as compared to changing the bed slope. As shown below, the data of the present observations are in agreement with those of Shields. The capacity of the pump was Q=130 l/s, with an accuracy better than 1% for discharges larger than about 5 l/s, as used in the present experiments. The water quality was good, because suspended sediment was settled before starting tests, in order to provide excellent visibility of the sediment surface.

The important issue of the test facility employed was its flow straightener consisting of two 10 mm geo-textile mattresses inserted at the upstream channel end in order to inhibit flow concentrations, suppress local turbulence and reduce wavy surface patterns. In addition the sediment bed increased slightly at the transition between the rigid intake section and the fully developed flow zone over the sediment bed, such that there were no formations of local intake scour. All experiments were thus run under plane bed conditions, and the bed elevation could be estimated to about 0.5d₅₀. The sediment surface was measured with a so-called shoe-gauge having a 4 mm by 2 mm wide horizontal plate at its base, whereas the water surface was read with a conventional point gauge, typically to ±0.5 mm, depending on local turbulence.

Six different sediments were considered, three of which were nearly uniform with a grain size 3.30, 4.80 and 0.55 mm, and three mixtures, with d₅₀=5.30, 1.20 (1.10 for small channel), 3.10 mm, and s=(d₈₄/d₁₆)^1/2=1.43, 1.80 and 2.15, respectively. The uniform sediment with d₅₀=3.30 mm had a density of r_s=1.42 t/m³, whereas the remainder had standard density 2.65 t/m³. The plastic sediment originated from a circular extruder, whereas the rest was from Swiss rivers with a typical ellipsoidal shape. The details of the granulometry and the tests conducted are available (Hager and Oliveto 2000) and a review of the most important test parameters is provided in Table 1. There, F_o=V_o/(gh_o)^1/2 and R_o=4h_oV_o/n are the Froude and Reynolds numbers in the approach channel. For mixtures the surface grain distribution was almost uniform by using a special grading mechanism, which allowed a vertical setting accuracy of 0.50d₅₀. In total, 60 inception series each with 5 to 7 experiments were conducted.

Table 1    Test conditions

4    EXPERIMENTAL RESULTS

Figure 1 shows the ratio of observed (subscript exp) and according to (6) to (8) computed (subscript com) densimetric particle Froude numbers F_i=F_dexp/F_dcom as a function of measured inception Froude numbers. It may be seen that all the data are confined within a ±10% band, as are the original Shields tests. Most of the present data were located in the transition regime because of limitations of the present facilities. For D_*<10, very small or very light particles would have to be used, with problems of water visibility. For D_*>150 or so, one had to take large or heavy particles, with limitations in the flow depth because of discharge capacity. These domains have been sufficiently checked in the past, however.

The data for sediment mixtures having a non-uniformity of larger than, say s =1.3 departed from the original equations (6) to (8). The larger s was, the smaller resulted F_i values. All data were thus corrected for non-uniformity with F_dm=F_di/s^1/3 instead of F_di according to (6) to (8), including those with 1<s <1.3. Mixtures used for experimentation were natural, and the data include sets up to s =2.15. Additional tests are necessary for s >2.5, say.

Fig. 1    F_i as Function of measured F_di; (△) d₅₀=4.80, s =1.25; (–) d₅₀ = 0.55 mm, s = 1.37; (´) d₅₀ = 5.30 mm, s = 1.43; (¡) d₅₀ = 1.20 mm, s =1.80; (¨) d₅₀ = 1.10 mm, s =1.70 (B = 0.5 m); (*) d₅₀ = 3.10 mm, s = 2.15.

5    APPLICATION OF RESULTS

Equations (6) to (8) exhibit interesting properties which are explored first. The dimensionless sediment size can also be expressed as D_*=d₅₀/d_R, with d_R=(n²/g’)^1/3 as a reference size, that depends exclusively on fluid viscosity and relative sediment density. Using d_R instead of d₅₀, the effect of the mean grain size may be discussed. Instead of (6) to (8), one may this write

D_*£10           s^-1/3V_i /(g’d_R)^1/2= 2.33 (d₅₀ /d_R)^1/12 (R_h /d_R)^1/6                          (9)

10<D_*<150          s^-1/3V_i /(g’d_R)^1/2= 1.08 (d₅₀ /d_R)^5/12 (R_h /d_R)^1/6                             (10)

D_*  ³150              s^-1/3V_i /(g’d_R)^1/2= 1.65 (d₅₀ /d_R)^1/3 (R_h /d_R)^1/6                  (11)

Except for the effect of d_R, the right hand side contains only mean sediment size and relative flow depth, whereas the left hand side incorporates the effects of sediment mixture, relative gravity and inception velocity. For all values of D_*, the modified Shields parameter s^-1/3 V_i /(g’d_R)^1/2 as also the relative flow depth (R_h /d_R)^1/6 remain unchanged. The effect of relative grain size d₅₀ /d_R thus is small for the viscous, strong in the transitional, and moderate in the fully rough regime. This result is astonishing, given that the effect of relative size is strong in the transition and not in the fully rough regime. Because of normalization, (11) contains the viscous effect but it is easily realized that d_R can be eliminated to result in

D_* ³150                s^-1/3 V_i /(g’³d₅₀²R_h)^1/6= 1.65                         (12)

Further, at least for this important case in field application, one might define an inception number I=(s^-1/3/1.65) (V_i/(g’³d₅₀²R_h)^1/6) and require that I£1 for no bed movement, whereas I>1 would result in bed movement. Such an inception number could be extended to 2D- computations, to verify whether a sediment bed remains locally stable or unstable. From (9) to (11) it is also obvious that the present procedure allows a straightforward determination of inception conditions that depends only on parameters known at design stage, whereas the standard procedure involves an estimation of parameters that are not readily available.

6    EXAMPLES

The previous relationships can be tested for typical applications in fluvial hydraulics. Consider standard conditions with a two-phase water-sediment flow, for which n=10^-6 m²/s at 20°C water temperature, r=1.0 t/m³ for water and r_s=2.65 t/m³ for usual sand, gravel or rock. The reference grain size diameter then is d_R=(n²/g’)^1/3=3.95´10^-5 m. Let us consider three cases with uniform sediment (s=1) where d₅₀=0.01 mm, 1 mm and 100 mm, for which D_*=0.253, 25.3 and 2530, respectively. Then, the inception velocities of sediment transport are for the R_h/d_R=10⁴, respectively, V_i=0.24 m/s, 0.49 m/s, and 2.64 m/s.

If the sediment would be non-uniform with s =2.0, the last case would give V_i=3.33 m/s instead of 2.64 m/s. For the last case also, one might determine the inception number to I=1, i.e. the critical inception number. From these three examples it may be noted that the velocity at inception of sediment transport is by no means linearly increasing with median grain size. Also, the effect of viscosity can be important in a hydraulic model, but not in prototype. The present analysis in a way reduces the mystery of Shields diagram, yet the legend of Shields to overcome, as addressed by Buffington (1999), needs considerable more investigations.

The limitations of this analysis may be summarized as follows:

(1) Sediments in the usual domain where the Shields diagram applies, thus with a relative density of approximately 0.1<(r_s-r)/r<7, and 0.2<D_*<200;

(2) Almost plane bed conditions, not too far away from uniform flow;

(3) Almost straight rectangular channel geometry, and a nearly horizontal sediment bed;

(4) Relative flow depth h_o/d₅₀>10; and

(5) Almost uniform distribution of sediment mixture, without local size accumulations.

7    CONCLUSIONS

An important issue in sediment hydraulics was addressed. The conventional Shields condition for inception of sediment transport was examined for flows close to uniform flows. By subdividing the Shields diagram into three portions, corresponding approximately to the viscous, the transition, and the fully rough regimes, and by using both Manning’s and Strickler’s equation for uniform flow and sediment roughness, respectively, a general equation for inception of sediment transport may be derived, containing exclusively basic hydraulic parameters. These include cross-sectional velocity, gravitational acceleration, kinematic viscosity, grain size, relative density of the two-phase flow and hydraulic radius. Using systematic experiments for sediment mixtures, one may define a mixture densimetric particle Froude number F_dm=V/(g’d₅₀) s ^-1/3 which accounts for the original Shields curve.

The resulting equations may be expanded using the concept of reference diameter d_R that accounts exclusively for viscosity and gravity. The normalized relations exhibit a maximum effect of relative grain size in the transition, instead of in the rough regime. Further, an inception number for the rough regime was defined that has a scaling length equal to (d₅₀²R_h)^1/3. The results are illustrated with computational examples. 

Acknowledgements

This project was a team work between University of Basilicata, Italy, and ETH Zurich.

References

Buffington, J.M. (1999). The legend of A.F. Shields. Journal of Hydraulic Engineering 125(4): 376-387.

Garcia, M.H. (2000). Discussion to The legend of A.F. Shields, by J.M. Buffington. Journal of Hydraulic Engineering 126(9): 718-720.

Graf, W.H., Altinakar, M.S. (1996). Hydraulique fluviale. Presses Polytechniques et Universitaires Romandes: Lausanne (in French).

Hager, W.H., Del Giudice, G. (2000). Discussion to Movable bed roughness in alluvial channels. Journal of Hydraulic Engineering accepted.

Hager, W.H., Oliveto, G. (2000). Shields condition in bridge hydraulics. Journal of Hydraulic Engineering submitted.

Kennedy, J.F. (1995). The Albert Shields story. Journal of Hydraulic Engineering 121(11): 766-772.

Patel, P.L., Ranga Raju, K.G. (1999). Critical tractive stress of nonuniform sediments. Journal of Hydraulic Research 37(1): 39-58.

Przedwojski, B., Blazejewski, R., Pilarczyk, K.W. (1995). River training techniques. Balkema: Rotterdam.

Shields, A.F. (1936). Anwendung der Aehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. Mitteilung 26. Preussische Versuchsanstalt für Wasserbau und Schiffbau: Berlin (in German).