Stability of bottom armoring under the attack of solitary waves
Creators: name: Naheer, Ehud
Publisher: name: California Institute of Technology
Year: 1977

description: An empirical relationship is presented for the incipient motion of bottom material under solitary waves. Two special cases of bottom material are considered: particles of arbitrary shape, and isolated sphere resting on top of a bed of tightly packed spheres. 

The amount of motion in the bed of particles of arbitrary shape is shown to depend on a dimensionless shear stress, similar to the Shields parameter. The mean resistance coefficient used in estimating this parameter is derived from considerations of energy dissipation, and is obtained from measurements of the attenuation of waves along a channel. A theoretical expression for the mean resistance coefficient is developed for the case of laminar flow from the linearized boundary layer equations and is verified by experiments. 

For the case of a single sphere resting on top of a bed of spheres, the analysis is based on the hypothesis that at incipient motion the hydrodynamic moments which tend to remove the sphere are equal to the restoring moment due to gravity which tends to keep it in its place. It is shown that the estimation of the hydrodynamic forces, based on an approach similar to the so-called "Morison's formula", in which the drag, lift, and inertia coefficients are independent of each other, is inaccurate. Alternatively, a single coefficient incorporating both drag, inertia, and lift effects is employed. Approximate values of this coefficient are described by an empirical relationship which is obtained from the experimental results. 

A review of existing theories of the solitary wave is presented and an experimental study is conducted in order to determine which theory should be used in the theoretical analysis of the incipient motion of bottom material. 

Experiments were conducted in the laboratory in order to determine the mean resistance coefficient of the bottom under solitary waves, and in order to obtain a relationship defining the incipient motion of bottom material. All the experiments were conducted in a wave tank 40 m long, 110 cm wide with water depths varying from 7 cm to 42 cm. The mean resistance coefficient was obtained from measurements of the attenuation of waves along an 18 m section of the wave tank. Experiments were conducted with a smooth bottom and with the bottom roughened with a layer of rock. The incipient motion of particles of arbitrary shape was studied by measuring the amount of motion in a 91 cm x 50 cm section covered with a 15.9 mm thick layer of material. The materials used had different densities and mean diameters. The incipient motion of spheres was observed for spheres of different diameters and densities placed on a bed of tightly packed spheres. The experiments were conducted with various water depths, and with wave height-to-water depth ratios varying from small values up to that for breaking of the wave. 

It was found that: (a) The theories of Boussinesq (1872) and McCowan (1891) describe the solitary wave fairly accurately. However, the differences between these theories are large when used to predict the forces which are exerted on objects on the bottom, and it was not established which theory describes these forces better. (b) The mean resistance coefficient for a rough turbulent flow under solitary waves can be described as a function of D_s, h, and H, where D_s is the mean diameter of the roughness particles, h is the water depth, and H is the wave height. (c) Small errors in the determination of the dimensionless shear stress for incipient motion of rocks result in large errors in the evaluation of the diameter of the rock required for incipient motion. However, it was found that the empirical relationship for the incipient motion of spheres can be used to determine the size of rock of arbitrary shape for incipient motion under a given wave, provided the angle of friction of the rock can be determined accurately.
description: The writer wishes to express his gratitude to his thesis advisor, Professor Fredric Raichlen, who suggested this research problem and offered his advice throughout every phase of the investigation. The advice and encouragement of Professors Vito A. Vanoni, Norman H. Brooks, and Peter S. Eagleson are also deeply appreciated. 

The writer also wishes to thank Dr. Robert C. Y. Koh and Dr. Sasson R. Somekh for the long discussions which helped in developing the procedures used in the experimental investigation. 

A special thanks is owed to Mr. Elton F. Daly, supervisor of the shop and laboratory, whose assistance during the design, construction, and maintenance of the experimental equipment made it possible to solve almost any problem in the laboratory phase of the investigation. Thanks are also due to Mr. Joseph J. Fontana, Mr. Robert Shultz, Mr. William G. Stone, Mr. Robert L. Greenway, and Mr. Carl A. Green who assisted in the construction and maintenance of the experimental equipment; to Mr. Walter Beckmann and Miss Sally Weaks, who helped in reducing the experimental data; to Mr. Gregory Gartrell and Mr. Philip J. W. Roberts, who reviewed the original manuscript, and to Mrs. Joan L. Mathews, Mrs. Shirley A. Hughes, and Mrs. Linda Rorem who typed it; and to Mr. David Byrum who performed the drafting of the figures appearing in this manuscript. 

The writer also wishes to thank the California Institute of Technology for financial assistance and for providing the facilities for this study. The experiments were conducted in the W. M. Keck Laboratory of Hydraulics and Water Resources. Thanks are also due to the Humanities Fund, Inc., for the Boris Bakhmeteff Research Fellowship which was awarded to the writer in 1972. The research was supported by National Science Foundation Grants GK-3l8D2X, ENG7l-D2367 AD3, and ENG75-l5786 AD1. 

The deepest gratitude is expressed by the writer to his wife, Daphna, for her understanding, patience, and encouragement during the period of his graduate study. 

This report was submitted by the writer on May 5, 1976 as a thesis with the same title to the California Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy; it is reproduced here unchanged.
description: Submitted - KH_R_34.pdf