BED LOAD TRANSPORT CAPACITY AND BED RESISTANCE IN EXCLUDER TUNNELS

Friedrich Schöberl

Prof. Dr., Innsbruck University

Institut für Wasserbau

Technikerstr. 13, A 6020 Innsbruck

Tel: ++43 512 507 6950

Fax ++43 512 507 2912

E-mail: f.schoe@uibk.ac.at

Abstract: Excluder tunnels are devices to effectively by-pass sediments to the tail water reach of water intakes. Inside the tunnel coarse sediments form a particular bed layer dominating the whole transport mechanism. The investigation tries to specify relations for different parameters of the bed and its roughness. The derived equations enable an improved basis for the hydraulic design of excluders  

Keywords: by-pass structure of coarse sediments, boundary layer development, high energy gradient flow, high transport capacity, moving bed layer, bed roughness

1    INTRODUCTION

Since the operation of water intakes is seriously handicapped by sediments, different measures are taken to prevent the entry of coarse material into the system. Very effective means in this respect are excluder tunnels situated below or beside the actual diversion structure. The preferred location of these tunnels accounts for the concentration of coarse sediments in the vicinity of the bed. In alpine rivers this facility frequently is combined with other sediment rejecting measures, each of them working in a specific range of flow. In any case the transport capacity of the excluder must be sufficient to convey the advancing bed load completely down to the tail water. Flow conditions in excluders are characterised by following features:

l         high energy gradient of flow

l         dominating non-uniformity with developing bed- and wall-boundary layer

l         specific transport regime due to the slightly adverse bed slope within the conduit

l         interrelation between bed roughness and bed load transport capacity

With regard to these complex conditions, the hydraulics of excluder tunnels differs from flow processes in long conduits or channels. Early approaches were mainly based on conventional transport relations, Garde (1976). Direct investigations of the transport mechanisms were performed by Kley (1988). According to this analysis the equivalent roughness of the bed ks becomes directly proportional to the acting bed shear stress. Main focus lies on the determination of the maximum flow velocity along the tunnel. For the estimation of hydraulic relevant magnitudes a set of equations is proposed which has to be simultaneously solved. Some open questions remain about the range of validity particularly with regard to the bed roughness and the geometrical features of the sediment bed. The relations developed in the following are mainly based on experimental data of Kley and on additional tests performed in a similar set up at the IWI (Institute of Water Resources Engineering Innsbruck ).

2    TRANSPORT CHARACTERISTICS

Differences occur between high energy gradient flows of practically horizontal excluders and the flows in steep channels. Within the conduit the formation of bed forms is totally depressed and the transport is rather characterised by the emergence of a moving bed layer. While bed load equations developed to describe the conditions of the upper transport regime fail to reproduce the transport relation adequately a relation in the form of the Meyer Peter-Müller (MPM) formula has the option to become appropriate when taking into the account the specific flow resistance of the moving bed layer. Therefore the main concern centres on the behaviour and the properties of the moving bed layer. According to the observations the thickness of the layer grows with increasing bed shear stress and furthermore is strongly related to the transport rate. The effective transport mechanism breaks only down for low energy gradients S < 0,02 and small transport concentrations c_s<0,005, where  and q_b specific bed load rate (m³/s,m), q_s specific discharge related to the bed. Of dominating influence is the self-adjustment of the bed roughness dependent on the layer thickness and the applied shear stress. As already observed by Kley (1988) the flow conditions along the layer are primary determined by the thickness of the moving layer and not by the property of the mixture i.e. the grain sizes of the transported sediments. The same holds for the transport capacity. On the other hand one has to consider that as soon as conditions approach towards the state of incipient motion the mechanism abruptly transforms to the normal transport behaviour with rising bed roughness. The transport capacity can be reproduced by the MPM-formula considering two facts:

l         due to the bed layer formation the effective diameter of the transported bed load differs from the average diameter of the bed material and with reference to the data the effective diameter d_e is dependent on the moving layer thickness d_s and approximately proportional to ks, the equivalent roughness of the bed.

l         the lower resistance of the moving layer increases the proportional coefficient of the original equation by a factor of 1,5.

In the case of high energy gradients and low bed slopes the equation writes in dimensionless terms

                                        (1)

 dimensionless bed load rate and

 dimensionless shear stress and

 dimensionless critical shear stress

where q_b is the volumetric specific transport rate (m³/s,m), d_e the transport effective grain-size, t bed shear stress, r_s density of sediment grains, r_w density of water,
g acceleration due to gravity.

According to Kley the thickness of the moving layer d_s and its roughness ks are interrelated magnitudes in the form , whereby ks is given in dimensional correct form by

                                (2)

The ks-expression of Kley agrees with data only for the highest level of shear stress and low transport concentrations and cannot be generally used for the determination of d_e. A revised analysis suggests

                             (3)

additional parameter to Equ. (1): d_s thickness of the bed moving layer and n viscosity of fluid. Fig. 1 compares the bed load discharge of the original MPM equation and the modified MPM relation for the moving layer regime in terms of non-dimensional magnitudes

3    LONGITUDINAL BED FORMATION AND BOUNDARY LAYER DEVELOPMENT

The inlet region of the excluder is characterised by non-uniform flow, see Fig.2. Turbulent boundary layers are growing along the bed and the walls, indicating an acceleration of flow. In balance with the rise of velocity the bed level increases simultaneously. The boundary development causes a significant change of the velocity distribution. At the entrance we find a constant velocity predominantly across the whole flow depth, steeply dropping down only in the vicinity of the boundaries. Further downstream the distribution follows a pronounced logarithmic shape evidently covering the whole cross-section. From this point downwards fully developed conditions can be considered. The distance where the velocity profiles of the upper wall and of the bed meet to form a combined maximum is hydraulically dependent on the flow depth. Since the bed creates the higher roughness compared to the walls, the bed boundary layer ds becomes the dominant parameter. In general the side-wall effects are small and can be ignored so that the hydraulic radius of the bed R_s becomes identical with the bed boundary layer thickness ds,

where  and F_s flow area related to the sediment bed according to the Einstein-subdivision concept of flow areas, b width of the sediment bed. As evidenced by data the distance x_e for fully established flow is directly related to R_s, see Fig. 3, resulting in a proportionality of

Since the shear stress along the bed must be constant for a continuous transport of bed load, the bed level has to adjust to the change of the velocity profile by forming an adverse bed slope. The rise of the bed level in the non-uniform region of the conduit shows equally a direct proportionality to R_s with , where w_e is the total increment of bed level within the inlet region, see Fig. 2.

4    BED ROUGHNESS AND FLOW RESISTANCE

The evolving bed roughness is directly interrelated with the grain transport characteristics. For the experimental data covering the conditions: 0,005<c_s<0,15 and 0,02<S<0,26 the equivalent roughness of the moving bed layer was recalculated by applying the Colebrook-White law. Although the flow in the front of the inlet region is non-uniform, the additional resistance due to this effect is small. So it is suitable to deduce the relation for established conditions which prevail in the back part of the excluder. Referring to the regime of a moving bed layer, roughness of bed layer becomes independent on grain size of sediments, see chapter 2. With reference to equation 3 an approximation for ks can be obtained by equalling ks and d_e, the effective magnitude governing the transport capacity. Although the level of magnitude is represented well, a certain scatter establishes for constant shear stresses and a clear additional dependency on the transport concentration c_s becomes evident. As deduced from the experiments, ks eventually becomes

                            (4)

where  

t bed shear stress, c_s bed load concentration, n fluid viscosity, r_w density of fluid, g¢ modified acceleration factor according Equ. (1). Taking into account that ks is a very sensitive parameter reacting progressively on smallest degrees of experimental inaccuracies, the performance of equation (4) depicted in fig. 4 describes an appropriate basis for calculation. Due to the additional dependency on c_s iterative computation becomes necessary, where as the simpler equation (3) can act as an initial value.

5    CONCLUSIONS

Transport and morphological status in excluder tunnels are hydraulically characterised by high pressure gradients due to the shortage of conduit. Transport concentrations cs>0,005 and energy slopes S>0,02 create a particular transport regime with a moving bed layer and its evolving roughness ks dominates and determines all other features: developing length and thickness of the bed boundary layer, amount of bed level increase and at last the resulting flow depth.

Experimental data shows clear evidence on the adjustment process of the bed and enables the formulation of simple equations of the geometrical interrelations. The roughness ks of the moving bed layer proves to be dependent on shear stress and the transport concentration as well. The specification of this key parameter establishes an adequate computation of flow and transport capacity.

References

Garde R.J.,Panda P.K.:Use of sediment transport concepts in design of tunnel-type sediment excluders, ICID Bulletin, Vol. 25 No. 2, 1976.

Kley G.M., Zur Bemessung von Geschiebeabzügen, Mitteilungen des Instituts für Wasserbau und Kulturtechnik, Universität Karsruhe, Heft 177, 1988.

Raudkivi A.J., Sedimentation – Exclusion and Removal of Sediment from Diverted Water, IAHR Hydraulic Structures Design Manual, No. 6, A.A. Balkema, Rotterdam, Brookfield, 1993.

Rickenmann D., Bedload transport capacity of slurry flows at steep slopes, Mitteilungen der Versuchsanstalt für Wasserbau, ETH Zürich, Heft 103, 1990.

Fig. 1    Comparison Meyer Peter-Müller formula and modified formula for moving layer regime

Fig. 2    Schematic longitudinal- and cross-section of an excluder

Fig. 3    Developing length x_e of flow establishment in relation to R_s

Fig. 4    Equivalent bed roughness ks referring to equation (4) versus
experimentally based results