A NEW Roughness Estimation Technique for granular-bed FLOW RESISTANCE

Graeme M. Smart

National Institute of Water & Atmospheric Research (NIWA)

Box 8602, Christchurch, New Zealand.

Tel.: +64 (3) 3437851, fax: +64 (3) 3485548, E-mail: g.smart@niwa.cri.nz

Abstract: For rough beds, the parameter of depth or hydraulic radius is difficult to define or measure. Volumetric hydraulic radius R is shown to be more suitable for use with the 2D bed shear stress equation. An investigation of the relation between R and depth when water level is below the tops of bed particles was carried out using digital elevation models (DEMs) of actual channel bed surfaces. For the water worked beds, a simple relationship was found and surface standard deviation d_z was one third of the distance from mean bed level to the tops of roughness elements. A simple sand spreading technique is proposed for measuring R and d_z.

Keywords: flow resistance, high relative roughness, hydraulic radius, in-situ roughness

1    INTRODUCTION

Flow resistance formulae relate flow velocity, hydraulic depth, slope and a roughness factor. However, when the water depth is similar to the size of bed protrusions, there is no standard definition for a roughness length scale or for the position on the bed from which to measure depth. Common techniques for determining the roughness of granular alluvial channels, such as grain-size sieving or Wolman (1954) sampling, require the bed to be disturbed.  As a consequence, the resulting roughness scale does not reflect the in-situ roughness of the bed surface which, for non-uniform grain-size distributions, will vary depending on grain orientation, arrangement and packing. A second problem arises in determining depth or hydraulic radius, as the bed datum is unclear and wetted perimeter becomes longer the finer the scale at which the bed is examined.  The conventional definition of hydraulic radius (flow area divided by wetted perimeter) is unfortunately a fractal that also depends on the bed roughness.

With regards to the roughness of the bed surface, it was recently shown that the standard deviation of bed surface elevations gives a better indicator of hydraulic roughness than grain size based measures (Aberle et al, 1999). If this approach is to be used, we require a technique to determine the standard deviation of a granular bed surface. Available techniques include laser altimetry, aerial photogrammetry and mechanical profilers as described by Smart et al. (2001).

2    BACKGROUND

In steady 2-D flow the downslope component of the weight of water is balanced by a shear force transferred to the channel boundary. Bed shear stress is given by the equation:

t_o = rgRS                                 (1)

where r is the density of water, g is gravitational force per unit mass, S the  channel slope and R is variously taken as a flow depth or hydraulic radius. From (1) it can be seen that S gives the downslope component of the weight of water and r g R is the total weight of overlying water per unit area (shear area is measured in the plane of the bed). However rg is the specific weight of water (weight/volume) so R of (1) is the volume of water per unit plane area of the bed. We call this definition of R the volumetric hydraulic radius.

For a bed with uniform properties in the flow direction, R is the area of water per unit width of the bed:

R = A_f /B                                   (2)

where A_f is the area of flow over a representative channel width B as shown in Fig. 1.

For situations where the water surface covers the tops of bed roughness elements it can be seen from Fig. 1 that the volumetric hydraulic radius R is the distance from mean bed level to the water surface, H. To determine R it is necessary to know the elevation of the mean bed level. Where the water surface does not cover the tops of bed particles, R is no longer equivalent to H as shown in Fig 2.

3    INVESTIGATIONS

An investigation of the relation between R and H in the case where water level is below the tops of bed particles, was carried out using digital elevation models (DEMs) of actual channel bed surfaces. The DEMs were collected by surveying laboratory and field channels with median grain sizes ranging from 4 mm to 0.4 m and d₉₀ grain sizes ranging from 7 mm to 0.8 m. The surveying techniques are described by Smart et al. (2001). With one exception (the Okarito River) the channel beds were placed by the action of water and included various degrees of form roughness as well as grain roughness. In order to determine bed properties in a plane parallel to the water surface slope, the raw DEMs were detrended by subtracting a trend surface which was the bi-cubic spline fit (Press et al, 1988) to the average of elevations lying within circles of diameter of 2.5 d₉₀.

The detrended bed surfaces were analysed as follows:

•  The standard deviation of elevations in the downstream direction was calculated. This parameter was termed d_z.

• Mean bed level was at the average elevation of the detrended DEM.

•  The volume of water per unit area of bed (R) was calculated for different depths of water (H) within the bed matrix with H measured from mean bed level (Fig.2).

4    RESULTS AND ANALYSIS

The distribution of bed surface elevations was found to be approximately normally distributed about mean bed level.

Thompson and Campbell (1979) and Bathurst et al. (1981) related flow resistance to the fraction of cross-section bed area blocked by roughness elements. Bathurst concluded it was not convenient to measure this area directly, however, with DEM representations of channel beds this is now straightforward. Figure 3 indicates the blocking of flow at different water levels for three of the channel beds. The blocking ratio shown is the ratio of solid volume (bed particles) to the total volume (bed particles plus water) at different heights above minimum depth level. The three beds display similar shaped curves. This approach requires knowledge of the elevation at which free surface water vanishes as the water level becomes lower and lower. The minimum water depth is shown on Fig. 2 and its position can be determined from curves of water volume versus water level derived from the DEMs.  For the three beds shown the distance of the minimum depth from mean bed level, H_o, varies from 10.6 mm to 0.87 m (see legend on Fig. 3.)

However, the data of Fig. 3 can be simplified by plotting parameters relative to water surface level rather than the more elusive minimum depth. This is shown in Fig. 4 for all seven channel beds investigated. The data appear to collapse onto a single curve given by the equation:

H / R = 1 ¨C (1+(H / 5d_z) ^-10                             (3)

A closer inspection of the insert on Fig. 4 shows that the Okarito River data deviate from this curve. The Okarito channel was bulldozed through a moraine and comprises boulders too large to be moved by flows in the channel. With (3) or Fig. 4, R can be calculated for water levels below the tops of bed particles if mean bed level and d_z are known. For the fully water-worked beds, the insert on Fig. 4 shows that H/R becomes unity when water depth is greater than about 3 d_z. In other words, the tops of bed elements lie about 3 standard deviations from the mean bed level, as could be expected for normally distributed data. Thus, in order to determine R from water level for beds of this type, all that is required is knowledge of the elevation of mean bed level and its distance below the tops of bed roughness elements (H_t on Fig. 2).

5    PRACTICAL APPLICATION

Mean bed level can be determined (for granular roughness larger than sand particles) by spreading a known volume of sand to the level of the tops of bed roughness elements and measuring the area of spread. For particles larger than cobbles, rounded pebbles can be used in place of sand making the procedure more practical to apply underwater. By using this technique:

•  The distance from the tops level to mean bed level H_t, is the sand volume divided by the area of spread.

•  The standard deviation (for the water worked beds investigated) d_z = H_t / 3.

•  For water levels above the tops of bed particles, volumetric hydraulic radius R = D + H_t where D is water depth above the tops of bed particles (Fig. 1).

• For water levels below the tops, R can be calculated from H, d_z and H_t using (3) or Fig. 4.

6    CONCLUSIONS

 In the 2D shear stress equation the parameter of depth or hydraulic radius is often not clearly defined.

•  The use of volumetric hydraulic radius R in this equation is better from a theoretical and practical point of view.

•  Volumetric R can be calculated for a given water depth if mean bed level is known.

•  For the water worked beds studied, surface standard deviation was one third of the distance from mean bed level to the tops of the largest roughness elements.

• The position of mean bed level can be determined by a simple sand spreading technique.

•  Better measures of R and roughness will lead to more accurate flow resistance prediction.

Acknowledgements

The research was carried out under contract CO1X0014 from the Foundation for Research, Science and Technology (New Zealand). Ian McEwan and Jochen Aberle provided bed topography data. Murray Hicks provided suggestions on the manuscript.

References

Aberle, J., Dittrich, A., and Nestmann, F. (1999). ¡°Description of steep stream roughness with the standard deviations S.¡± In Fluvial Systems ¨C Processes, Functions and Management Proc., XXVII IAHR Biennial Congress, Graz, Austria, August 1999, on CDROM.

Bathurst, J.C., Li, R.M., and Simons, D.B. (1981). ¡°Resistance equation for large-scale roughness¡±. J. Hyd. Div., ASCE 107 (12), 1593-1613.

Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. (1988). Numerical Recipes in C. Cambridge University Press, Cambridge.

Smart G.M., Duncan MJ., Walsh J. (2001) ¡°Relatively Rough Flow Resistance Equations¡±. Journal of Hydraulic Engineering, ASCE, submitted.

Thompson S.M. and Campbell P.L. (1979) ¡°Hydraulics of a Large Channel Paved with Boulders¡±. Journal of Hydraulic Research, IAHR 17(4): 341-354.

Wolman, M.G., (1954). A method of sampling coarse river bed material. American Geophysical Union Transactions 35: 951-956.

Fig. 1  Cross section of channel bed, water level above the tops of bed particles,

R = H

Fig. 2  Cross section of channel bed, water level below the tops of bed particles.

R ¹ H.

Fig. 3  Percent of bed area blocked by roughness elements at different distances from the bed

Fig. 4    Relation between volumetric hydraulic radius R and depth from mean bed level H, for channels with different surface roughnesses (of standard deviation d_z). The Okarito bed is not water worked