 Definition sketch for the shape of a self-formed channel in noncohesive alluvium.

WHAT IS THE ASPECT RATIO OF A SELF-FORMED ALLUVIAL STREAM?

Victor M. Ponce

San Diego State University, San Diego, California

24 September 2023

 ABSTRACT. The aspect ratio of a self-formed stream in noncohesive alluvium has been calculated using the Lane and others (1959) theory. The aspect ratio To /Yo is expressed in terms of the friction angle φ of the surrounding soil and the lift-to-drag force ratio β acting on each particle. The results agree very well with practical experience. An online calculator may be used to show the sensitivity of the calculated results to the input variables φ and β.

1.  INTRODUCTION

Alluvial streams are self-formed, i.e., their shape, characterized by depth and width, obeys the laws of mechanics, properly within the field of river mechanics. The ratio of width to depth is referred to as the aspect ratio of the cross-section. The study of alluvial river flow encompasses the calculation of three variables: (1) depth, (2) width, and (3) slope. Depth and slope may be calculated using the laws of fluid mechanics, reflected in the fundamental laws of conservation of mass, momentum, and energy (Chow, 1959; Ponce, 2014).

Momentum integrates the acting force in time, while energy integrates the same force in space; therefore, they are not independent. In steady flow in open channels, the equations of mass and energy are used to solve for the unknown flow variables (depth and slope). In unsteady flow, the equations of mass and momentum are used for the same purpose. This reality leaves the calculation of stream width out of fluid mechanics. The width would have to be calculated using principles of river mechanics.

Lane and others (1959) developed a mechanistic approach to calculate the width in a self-formed channel. The theory has been recently revisited by Ponce and Jiang (2020). While Lane and others' approach is limited to noncohesive materials under equilibrium conditions, it is a point-of-start in the nascent field of unsteady alluvial river mechanics.

The application of Lane's methodology may serve to throw additional light onto relevant issues of floods, flood damages and related sedimentation issues (ASCE, 1975; 2007). Moreover, in light of contemporary global climate change, the subject is now becoming increasingly important.

2.  LANE AND OTHERS' THEORY

The size of the cross-section for a stream channel in equilibrium, formed with noncohesive materials, is dependent on the maximum threshold discharge. Using principles of river mechanics, Lane and others (1959) showed that the shape of the equilibrium cross section is a function of the following variables: (1) the friction angle φ of the material forming the channel bed, and (2) the lift-to-drag force ratio β acting on a particle.

The main assumptions used by Lane and others in deriving their theory are the following:

1. The side slope of the channel, at or above the water surface, is equal to the friction angle φ of the noncohesive alluvial material (Fig. 1).

2. The particles at the channel boundary are at the condition of incipient motion. The lift and drag forces acting on each particle, together with the downward component of the gravity force, are balanced exactly by the frictional force developed between particles. The lift and drag forces are assumed to be proportional to the tractive force exerted on the channel bottom. The lift-to-drag force ratio is referred to as β.

3. The particles are held in place by virtue of their submerged weight resolved in a direction normal to the bed.

4. The tractive force acts in the direction of the flow and is equal to the weight of the column of water above the area on which the force acts.

5. The channel is assumed to be prismatic, with negligible secondary currents and the absence of backwater effects. Fig. 1  Definition sketch for the shape of a self-formed channel in noncohesive alluvium.

3.  FINDINGS

The derivation of the Lane and others (1959) theory has been extensively documented by Ponce and Jiang (2020): Section 3. In that report, Eq. 57, reproduced here as Eq. 1, is the equation derived by Lane and others for the width-to-depth ratio, or aspect ratio of a self-formed channel carved in noncohesive material under equilibrium conditions:

 To                2          1 + r ______  =  _________ (______) 1/2  cos-1 r    Yo             tan φ      1 - r (1)

in which To = channel top width, Yo = maximum (center) flow depth, φ = friction angle of the noncohesive material, β = lift-to-drag force ratio, and r = β tanφ.

Table 1 shows aspect ratios calculated using Eq, 1, for friction angle varying in the range 5 ≤ φ ≤ 45°, at intervals of 5°, and lift-to-drag-force ratio varying in the range -1 ≤ β ≤ 1, at intervals of 0.1. For a typical value of φ = 30° and β = 0.4 (Apperley, 1968), the aspect ratio is: To / Yo = 5.86. Likewise, for φ = 45° and β = 1:  To / Yo = 4.

 Table 1.  Value of To /Yo as a function of friction angle φ and lift-to-drag force ratio β. β Friction angle φ (degrees) 5 10 15 20 25 30 35 40 45 -1.0 34.728 16.591 10.447 7.292 5.320 3.920 2.814 1.809 0.001 -0.9 34.849 16.722 10.589 7.448 5.495 4.124 3.064 2.160 1.235 -0.8 34.970 16.851 10.728 7.598 5.660 4.310 3.284 2.438 1.665 -0.7 35.090 16.978 10.862 7.743 5.817 4.484 3.480 2.672 1.971 -0.6 35.210 17.103 10.994 7.882 5.966 4.645 3.658 2.874 2.214 -0.5 35.328 17.226 11.123 8.017 6.109 4.796 3.822 3.054 2.418 -0.4 35.446 17.348 11.248 8.148 6.245 4.939 3.973 3.216 2.595 -0.3 35.563 17.467 11.371 8.274 6.375 5.074 4.113 3.364 2.752 -0.2 35.679 17.585 11.492 8.397 6.500 5.202 4.245 3.500 2.894 -0.1 35.794 17.702 11.609 8.516 6.621 5.325 4.369 3.626 3.023 0.0 35.909 17.817 11.725 8.631 6.737 5.441 4.487 3.744 3.142 0.1 36.022 17.930 11.838 8.744 6.849 5.553 4.598 3.855 3.252 0.2 36.135 18.042 11.948 8.854 6.958 5.660 4.704 3.959 3.354 0.3 36.248 18.153 12.057 8.961 7.063 5.763 4.805 4.058 3.451 0.4 36.359 18.262 12.164 9.065 7.165 5.863 4.902 4.152 3.542 0.5 36.470 18.370 12.269 9.167 7.264 5.959 4.994 4.241 3.628 0.6 36.580 18.476 12.371 9.266 7.360 6.051 5.083 4.327 3.709 0.7 36.690 18.581 12.473 9.363 7.453 6.141 5.169 4.408 3.787 0.8 36.799 18.685 12.572 9.458 7.544 6.227 5.252 4.487 3.861 0.9 36.907 18.788 12.670 9.551 7.632 6.311 5.331 4.562 3.932 1.0 37.014 18.889 12.766 9.642 7.718 6.393 5.408 4.635 4.000

4.  ONLINE CALCULATION

Equation 1 may be solved with the aid of the online calculator ONLINE LANE SHAPE. For instance, for Yo = 1 m, φ = 30°, and β = 0.4, the answer is: To = 5.8627 m, which is in agreement with the result shown in Table 1.

[Click on top of image to expand]. 5.  CONCLUDING REMARKS

The aspect ratio of a self-formed stream in noncohesive alluvium has been calculated using the Lane and others (1959) theory. The aspect ratio To /Yo is expressed in terms of the friction angle φ of the material forming the channel bed and the lift-to-drag force ratio β acting on each particle. The results are seen to agree very well with practical experience. An online calculator may be used to show the sensitivity of the calculated results to the input variables φ and β.

REFERENCES

Apperley, L. W. 1968. The effect of turbulence on sediment entrainment. Thesis submitted to the School of Engineering, University of Auckland, Auckland, New Zealand, in partial fulfillment for the requirements for the degree of Doctor of Philosophy, January.

ASCE, 1975. Sedimentation Engineering. Manuals and Reports on Engineering Practice, Manual 54, Vito A. Vanoni, editor, New York.

ASCE, 2007. Sedimentation Engineering: Processes, Measurements, Modeling, and Practice. Manuals and Reports on Engineering Practice, Manual 110, Marcelo H. Garcia, editor, New York.

Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, Inc, New York.

Lane, E. W., P. N. Lin, and H. K. Liu. 1959. The most efficient stable channel for comparatively clean water in noncohesive material. Report CER-59HKL, Colorado State University, Fort Collins, Colorado, April.

Ponce, V. M., and S. Jiang. 2020. The equilibrium shape of self-formed channels in noncohesive alluvium. Online report.
https://ponce.sdsu.edu/the_equilibrium_shape_of_self_formed_channels.html

Ponce, V. M. 2014. Fundamentals of Open-channel hydraulics. Online textbook.
https://ponce.sdsu.edu/openchannel/index.html

 230924