INSTABILITY OF DEVELOPED ALTERNATE BAR ON NON

¹Department of Civil Engineering, Faculty of Engineering, The University of Tokushima,

Abstract: In flume tests, it was found that there are some hydraulic conditions in which alternate bars can hardly attain their equilibrium; developed bars fade with decrease of wave height or upstream bars catch up with downstream ones. Such alternate bars were not observed in flume tests using uniform sediment. It is supposed that longitudinal and transverse sediment sorting is responsible for such instability of developed alternate bars. In the present study, instability of developed alternate bars on non-uniform sediment bed is discussed by means of numerical analysis and flume tests. The results suggest that alternate bars formed on non-uniform sediment bed do not approach an equilibrium state under some hydraulic conditions. Time lag between the time necessary for accomplishing equilibrium bed geometry and the time for sediment sorting adjusted to the bed geometry is responsible for instability and unsteadiness of alternate bar bed. Furthermore, we found that developed alternate bars formed on non-uniform sediment bed become unstable under the hydraulic conditions where the time scale of sediment sorting development is almost the same as that of alternate bar formation. The hydraulic conditions in which bars become unstable are discussed by means of linear bed stability analysis. The results suggest that the disturbance amplitude of mean diameter of sediment may be a parameter for predicting of bar bed instability.

Keywords: non-uniform sediment, sediment sorting, alternate bar, numerical analysis, linear bed stability analysis

1 INTRODUCTION

Previous theoretical and numerical studies of equilibrium geometric characteristics of alternate bars suggest that local transverse bed slope affects the movement of bed sediment so as to suppress the growth and equilibrium value of wave height (Fukuoka et al, 1985; M. Colombini et al, 1987; Shimizu, 1991; Kuroki et al, 1992). In these studies, uniform sediment was used. In our previous work (Takebayashi et al, 1998), the geometric and migration characteristics of alternate bars formed on non-uniform sediment bed were investigated by means of flume experiments and numerical analysis. It was shown that longitudinal and transverse sediment sorting affects the geometric and migration characteristics of alternate bars. Furthermore, we found that there are some hydraulic conditions in which it is difficult to obtain equilibrium alternate bars in the flume tests. Such alternate bars are not observed in flume tests using uniform sediment for bed material. It is supposed that longitudinal and transverse sediment sorting is responsible for such instability of developed alternate bars. In the present work, instability of developed alternate bars on non-uniform sediment bed is discussed based on the results of numerical analysis, linear bed stability analysis and flume tests.

		Non-dimensional shear stress	Width-depth ratio	Thickness of exchange layer
Case 1	Simulation	0.097	19.5	Maximum diameter
Case 2	Simulation	0.097	19.5	Maximum diameter×4
Case 3	Simulation	0.097	19.5	Maximum diameter×1/4
Case 4	Simulation	0.097	13.0	Maximum diameter
Case 5	Simulation	0.097	14.9	Maximum diameter
Case 6	Simulation	0.097	39.0	Maximum diameter
Case 7	Simulation	0.097	51.9	Maximum diameter
Case 8	Experiment	0.085	22.3	-

2 ANALYSIS METHOD AND EXPERIMENTAL PROCEDURE

2.1 Numerical analysis

The analyzed flow domain is a rectangular, straight open channel with a bed slope of 1/90. Water and sediment are supplied at a constant rate from the entrance. Depth integrated conservation equations of mass and momentum are employed for water. In order to evaluate sediment budget vectors, the velocity components at the bed surface are evaluated using the curvature of streamline (Engelund, F., 1974). The sediment transport rate is evaluated by means of the Ashida-Michiue formula, and the influence of local bed inclination on sediment transport rate and its direction are estimated (Ashida, K., Egashira, S. and B. Y. Liu, 1991). Theory of exchange layer (Hirano, 1972) is applied to evaluate sediment grain size distribution. Grain size distribution is estimated using the following equation:

where f_mp is the concentration of sediment of size class p in the exchange layer, z_b is the bed elevation, E_m is the thickness of exchange layer, q_bxp and q_byp is the sediment transport rate of size class p in x and y directions, respectively, l is the porosity of bed sediment. The flow on alternate bars exhibits a mixed regime with sub- and super-critical regions depending on flow conditions. Hence the equations are discretized using the TVD-MacCormack scheme.

Initial bed geometry has a disturbance of h/3 high, B/5 wide and B long near the upstream end, where h and B are the water depth and the flume width, respectively. Several disturbance types are employed in the calculation. Geometry and location of bed disturbance, however, do not affect the phenomena discussed in this research. Hydraulic conditions used for the analysis are shown in Table 1. Cases 1 to 3 have the same conditions except for a thickness of exchange layer. The effect of sediment sorting on developed bar instability is discussed with regard to the result obtained from these conditions. Cases 4 to 7 have the same non-dimensional shear stress but a large difference in width-depth ratio. Non-uniform sediment, which has value of 1.93 in standard deviation and 1.09mm in mean diameter, is employed as bed material. Linear bed stability analysis predicts these hydraulic conditions should produce alternate bars.

2.2 Linear bed stability analysis

Basic equations the same as for numerical analysis are employed and linearized for bed stability analysis. The non-dimensional forms are as follows:

where k and l are the wave numbers in x and y directions, respectively, F is the Froude number, I is the bed slope, u and v are the velocity components in x and y directions, respectively, d_m is the mean diameter of sediment, d_p is the sediment diameter of size class p,

is the complex migration velocity, t_*p is the non-dimensional shear stress of size class p, t_*cp is the non-dimensional critical shear stress of size class p, m_k is the coefficient of dynamic friction, m_s is the coefficient of static friction, the subscript 0 denotes the value of normal flow, ^ denotes the non-dimensional amplitude of disturbance and n is the number of size class of bed material. nth order equation of

will be obtained under the condition where the determinant of coefficient matrix of Equation (2) is equal to 0. A value of

which is almost equivalent to the migration velocity of bars on uniform sediment bed using a sediment material having narrow standard deviation is chosen as the migration velocity of sand bars on non-uniform sediment bed from n solutions of

. Subsequent analytical procedure is the same as that used for uniform sediment bed (Kuroki, M. and Kishi, T., 1982).

2.3 Experimental procedure

Experiments were carried out under conditions of constant flow discharge and sediment feeding rate in a rectangular straight open channel, 12 m long and 30 cm wide. The initial flume bed was made up smoothly. The flume was fed with the same material as bed material from the upstream end. Experimental condition is shown in Table 1. Measurements were performed for bed elevation and surface bed material. Grain size of bed surface was investigated by collecting material from surface to depth of maximum grain size. Photographs were taken from the right side of the flume.

3 RESULTS AND DISCUSSION

A fading of alternate bars is shown in Figure 1. The circled alternate bar developed before 4000s. The wave height, however, decreases with time and the bar almost disappears at 5200s.

Figure 2 shows the fading of bars observed in a flume test. The photos show the trough of a bar. The trough is being filled with transported sediment over time and the bar finally disappears. It is considered that longitudinal and transverse sediment sorting is responsible for this instability of developed alternate bars. Equation (1) shows that a finite time will be spent for the development of sediment sorting on bars. A time lag, therefore, will be produced between the time when the bed geometry approaches to an equilibrium state and the time when sediment sorting is adjusted to the bed geometry. Our previous work (Takebayashi et al, 1998) suggests that sediment sorting affects the geometric and migration characteristics of alternate bars; wave height, wavelength and migration velocity of alternate bars formed on non-uniform sediment bed is lower, shorter and faster than those of bars formed on uniform sediment. Hence, the geometric and migration characteristics of bars change with time during the development of sediment sorting. Consequently, it is considered that development of sediment sorting causes a decrease of wave height after 4000s as shown in Figs. 1 and 2. Furthermore, change of migration velocity depending on development of sediment sorting causes a combination and collapse of bars. Two bars that have the same bed geometry are considered as follows. Sediment sorting is not developed on one bar that is located downstream from the other. On the other hand, sediment sorting is developed on the upstream bar. The migration velocity of the upstream bar is faster than that of the downstream one because sediment sorting makes the migration velocity of bars fast. As a result, the upstream bars catch up with the downstream one and the bars are combined.

Figure 3 shows the bed geometry in Cases 1 to 3. Periodical alternate bars are formed in Case 2, but the bars are irregular in Case 1. These results show that the instability of bars depends on the development of sediment sorting. Sediment sorting in Case 2 is not developed very much because of the thick exchange layer. As a result, bars in Case 2 become stable. A remarkable phenomenon is found in Case 3. Periodical alternate bars are formed in spite of a thin exchange layer. Sediment sorting adjusts to bar geometry quickly in Case 3, because the time scale of sediment sorting development is significantly shorter than the time scale of bar development. Consequently, the geometric and migration characteristics difference between the upstream and downstream bars are small. These results suggest that developed bars will become unstable under the condition that time scale of sediment sorting development is almost the same as that of alternate bar formation.

Finally, hydraulic conditions for instability of alternate bars are discussed focusing on the relation between width/depth ratio and the non-dimensional disturbance amplitude of mean diameter of bed material（

is the phase lag between

and

）. Figure 4 shows the results obtained from linear bed stability analysis. It is considered that a large value of

indicates the well development of sediment sorting. Solid and broken lines denote the value of

obtained by means of linear analysis. Black circles and white triangles show the hydraulic conditions where stable alternate bars are formed and developed bars become unstable in numerical analysis, respectively. Bars are judged to be unstable bars, when they disappear or combined.

agrees well with the instability of developed bars obtained by means of numerical analysis. This result suggests that

can be a parameter for prediction of bar bed instability. Furthermore, Fig. 4 suggests that developed bars should be unstable in a range of

(2.1＜

＜2.5～4.1).

Fig. 4 Non-dimensional disturbance amplitude of mean diameter of bed material
(E_m: thickness of exchange layer, d_max: maximum diameter of bed material)

4 CONCLUSION

Instability of developed alternate bars on non-uniform sediment bed was discussed by means of numerical analysis, linear bed stability analysis and flume tests. The results are summarized as follows: (1) Alternate bars formed on non-uniform sediment bed do not approach an equilibrium state under some hydraulic conditions. (2) Instability of developed alternate bars depends on the development of sediment sorting. Developed alternate bars formed on non-uniform sediment bed become unstable under the conditions where time scale of sediment sorting development is almost the same as that of alternate bar formation. (3) Disturbance amplitude of mean diameter can be a parameter for prediction of bar instability.

The authors wish to thank Mr. Angus Alexander McDonald, for his helpful suggestions about the English text. Mr. Y. Hasegawa, graduate student of Ritsumeikan University, assisted flume test and data processing. The authors are grateful to his supports.

Ashida, K., Egashira, S. and B. Y. Liu (1991). Numerical method on sediment sorting and bed variation in meander channels, Annual Journal of Hy. Eng. JSCE, Vol. 35, pp. 383-390 (in Japanese with English abstract).

Engelund, F., (1974). Flow and bed topography in channel bends, Jour. of Hy. Div. ASCE, Vol. 100, No. HY11.

Fukuoka, S. and Yamasaka, M. (1985). Equilibrium height of alternate bars based on non-linear relationships among bed profiles, flow and sediment discharge, Proc. of JSCE, No. 357, pp. 45-54 (in Japanese).

Hirano, M. (1972). Studies on variation and equilibrium state of a river composed of nonuniform material, Proc. of JSCE, No. 207, pp. 51-60 (in Japanese).

Kuroki, M. and Kishi, T. (1982). Study on Regime Criteria of River Morphrogy, Annual Journal of Hy. Eng. JSCE, Vol. 26, pp. 51-56 (in Japanese).

Kuroki, M., Ishii, K. and Itakura, T. (1992). Theoretical analysis on the wave height of alternating bars, Annual Journal of Hy. Eng. JSCE, Vol. 36, pp. 1-6 (in Japanese with English abstract).

M. Colombini, G. Seminara and M. Tubino (1987). Finite-amplitude alternate bars, J. Fluid Mech. vol. 181, pp. 213-232.

Shimizu, Y. (1991). A study on prediction of flows and bed deformations in alluvial streams, Thesis presented to Hokkaido University (in Japanese with English synopsis).

Takebayashi, H., Egashira S. and H. S. Jin (1998). Numerical simulation of alternate bar formation, Proc. of 7th ISRS and 2nd ISEH, pp.733-738.