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Chapter 7 - Establishing an empirical formula
7.6 Summary and conclusions
7.6.1 Summary
A dimensional analysis (§ 7.2) allowed for the identification of the main parameters influencing the
scour process. These are the overall energy slope S eall , , the water depth to channel width ratio
h m ⁄ B , the ratios V⋅ R h ⁄ g⋅
B 3 (replacing the Reynolds number that showed an important
correlation) and R c ⋅ h m ⁄ B 2 . The frequently used ratio V∗ ⁄ V and Fr d do not seem to play a
determinant role. An additional parameter was found to characterize the macro-roughness: the
spacing of the ribs e s . To make this ratio dimensionless it can be divided either by the channel
width B, the mean water depth h m or the radius of curvature R c .
In a first step, existing scour formulae were enhanced (§ 7.3.2). Table 7.6 gives a summary of the
obtained results. Since the formula of KIKKAWA ET AL. (1976) is quite frequently used in practice,
an effort was made to enhance its prediction capability for mountain rivers with a coarse bed by
adding additional parameters. The result is quite good but it is possible to get a better prediction
with other equations. BRIDGE (1976) gives the smallest error on the predicted average scour
depth. Therefore, this formula was enhanced too. By introducing two additional parameters,
related to the channel geometry, the correlation for the maximum scour depth can be increased to
0.82 and the predicted cross-section shape fits quite well to the measured one. The predicted values
fit well to the results obtained with scale model tests (Gurtnellen) and field data (Sacramento
River) (§ 7.5).
Since the observed cross-section is rather s-shaped than an exponential function (which is an
assumption of most existing formulae), an attempt was made to establish a scour formula based
on the shape of the cross-section (§ 7.3.3). It was assumed that the maximum bed slope in a radial
direction is limited to a maximum value corresponding to the friction angle of the bed material.
Different polynomial functions (Table 7.6) of the third and fifth degree were tested. A large set of
boundary conditions was analyzed, by fixing the cross-section slope at the outer and / or inner
bank and by admitting or omitting a symmetry point on the channel axis. The results range from
poor to good. The best result was obtained with a polynomial function of the third degree
(equation 7.20) obtaining a correlation of R 2 = 0.856 and fitting well to the measured cross-section.
Unfortunately this equation shows some lacking (underestimation of the scour depth) when
applied to additional laboratory and field data (§ 7.5 and Fig. 7.19).
Hoping to increase the precision of the predicted scour depth, an approach based on the similitude
and approximation theory of KLINE (1965) was explored (§ 7.3.4). Based on the most important
forces acting on a control volume in the cross-section, dimensionless ratios between the
different characteristics were derived. The advantage of this method is that the parameters
obtained in this way are based on the physical process. Unfortunately the result is not very satisfying.
The highest obtained correlations are of 0.72 and for the case with macro-roughness
R 2 < 0.5 .
page 188 / November 9, 2002
Wall roughness effects on flow and scouring