MECHANICS OF SEDIMENT MOVEMENT

Lecture notes for Short Course No. 3, sponsored by the Eastern Section of the Society of Economic Paleontologists and Mineralogists, and given in Providence, Rhode Island, March 13-14, 1984

Gerard V. Middleton, McMaster University John B. Southard, Massachusetts Institute of Technology

SECOND EDITION MPRCH 1984

Printed in U.S.A.

© Copyright, SEPM, 1984 All rights reserved.

IN ORDER TO FACILITATE THE MOST TIMELY PUBLICATION, SEPM SHORT COURSE NOTES ARE NOT SUBJECTED TO THE LEVEL OF REVIEW REQUIRED OF SEPM SPECIAL PUBLICATIONS.

i

PREFACE

As in the first edition, our purpose in writing these notes has been to present a discussion of a few topics central to a physical understanding of the mechanics of sediment movement. We assume that the reader has some calculus and elementary Newtonian physics, but not necessarily any prior experience in fluid mechanics. In such a brief treatment it has been necessary to select only a few topics for discussion. We have confined ourselves to unidirectional flows (excluding waves) of relatively small scale (excluding Coriolis effects) Even so, a large number of topics have been considered not at all or only in passing, including one of the most important problems in sediment mechanics: theories for the prediction of bed-load and suspended-load discharge. It seems to us more important to try to develop some physical insight about the elementary processes of sediment movement than to attempt to elaborate any comprehensive quantitative theories of sediment movement. We refer the reader to more detailed treatments of many of the topics covered here by Graf (1971), Raudkivi (1976), Vanoni (1977), and Yalin (1977). Also noteworthy is the magnificent review and analysis of the mechanics of development of sedimentary structures by Allen (1982) .

It is again our pleasure to dedicate these notes to Vito A. Vanoni, Professor of Hydraulics, Emeritus, and Norman H. Brooks, James Irvine Professor of Environmental and Civil Engineering, of the California Institute of Technology. Both of us, at different times in the now-distant past, spent many months at Calteci-i attempting to provide ourselves with a better basis for applying the science of sediment transport, and the techniques of study so well developed by hydraulic engineers, to geological problems. In this endeavor we were inspired by the guidance, hospitality, and interest on the part of Professors Vanoni and Brooks. We also owe a large debt of gratitude to Elton Daly, that master flume builder at Caltech, for his advice and services much beyond what we expected.

We extend special thanks to the present group of sedimentology graduate students at McMaster and MIT--Pedro Moreno Hentz, Roger Kuhnle, John Lambie, Marg Rutka, Joe Pozzobon, Doug Walker, Peter Wilcock, and Shenmin Zhang--for perceptive comments on style and substance of these notes, and for the tedious task of detecting typos. We're also indebted to the production line--Fran Doughty, Judith Stein, and Joan Hirschfeld--for going out of their way to get these notes ready for printing under the unreasonable time pressure brought about by the author's tardiness. G.V. Middleton J.B. Southard

iv

LIST OF COMMONLY USED SYMBOLS A

A0 A1 a1 a2 B B'

b C

CD CL C

C1

C2 D D

D0 d

d° E

F

Fr Fr FD FG FL

FR f

fo f

g g'

H H0

H0 HL

h

h0 h I

cross-sectional area normal to the flow; inverse of von KSrmn constant; area of ripple in streamwise profile constant in analysis of bed-form shape constant of integration in analysis of the law of the wall for smooth flow; constant in analysis of bed-form shape coefficient in analysis of incipient movement coefficient in analysis of incipient movement constant term in law of wall for smooth flow; Bingham number constant term in law of wall for fully rough flow distance between clefts along gravity-flow head fractional volume concentration of grains in a fluid; Chzy coefficient; cohesion; speed of sound drag coefficient lift coefficient celerity of surface gravity waves constant of integration; coefficient in analysis of incipient movement constant of integration; coefficient in analysis of incipient movement diameter nominal diameter dimensionless sediment diameter flow depth dimensionless flow depth energy energy per unit weight of fluid shear stress exerted by flow on saltation carpet or traction carpet Froude number Froude number based on reduced gravity g' drag force gravity force lift force resistance force function symbol; friction factor friction factor associated with lower surface of density current friction factor associated with upper surface of density current acceleration due to gravity reduced gravity, g' = g(p-p)/p or g' = g[ps-p)/pg] mean height of grain rise in saltation; height of bed forms specific head dimensionless ripple height head loss elevation above datum; elevation head; thickness of gravity-flow head elevation of channel bottom elevation of water surface height to overhang of turbidity-current head moment of inertia

V

K K1

L° M

constant in analysis of bed-form shape constant in analysis of incipient moVement porosity correction factor in sediment conservation equation, K1 = 11(1-x) constant in analysis of incipient movement7 constant in analysis of bed-form shape constant in analysis of bed-form shape equivalent sand roughness length dimension; characteristic length scaler bed-form spacing dimensionless sediment concentration, L = D/S dimensionless bed-form spacing mass dimension

m

mass

K2 K3 k5 L

N

dimensionless number of grains in motion per unit area and unit time n number of grains in motion per unit area and unit time p wetted perimeter; flow power normal stress transmitted by grain collisions fluid pressure; pore pressure p discharge Q q discharge per unit width volumetric bed-form transport rate per unit width qf volumetric sediment transport rate per unit width qs q5b volumetric bed-load transport rate per unit width R derivative of sediment transport rate with respect to bed elevation, R = dq5/dh Re Reynolds number Re* boundary Reynolds number, Re* = u*D/v hydraulic radius RH S slope, S = tanq; average spacing of particles S.F. Corey Shape Factor T time dimension; period of bursts; shear stress transmitted by grain collisions Tr passage time for a bed form past a station shear stress transmitted by both grain collisions and fluid t5 thickness of traction carpet U flow velocity averaged over a cross section, U = Q/A bed-form velocity UB grain velocity UG U0 dimensionless flow velocity Umax maximum or surface velocity surface velocity in open channel flow Us u instantaneous fluid velocity in the x direction time-average fluid velocity in the x direction u' instantaneous deviation from i, u' = u u* shear velocity, u* = (T0/p)112 u*c value of u* at threshold of sediment movement V fluid velocity, used in various ways; characteristic velocity scale y instantaneous fluid velocity in the y direction time-average fluid velocity in the y direction vi instantaneous deviation from y' = y w work WI submerged. weight of saltating grains per unit area of bed ,

vi

r

settling velocity (fall velocity) instantaneous fluid velocity in the z direction time-average fluid velocity in the z direction instantaneous deviation from W, w1 = w streamwise coordinate direction cross-stream coordinate direction normal to boundary roughness length displacement length dimensionless distance from boundary cross-stream coordinate parallel to boundary; Rouse number, z = w/i3Ku* angle of internal friction; angle of imbrication; ripple slip-face angle Shields parameter; coefficient relating and c, c = value of Shields parameter at threshold of sediment movement weight per unit volume (specific weight) of fluid, y = pg weight per unit volume (specific weight) of solid, y = p5g downchannel component of y submerged weight per unit volume of solid, y' = thickness of boundary layer thickness of viscous sublayer eddy diffusion coefficient (kinematic eddy viscosity) eddy diffusion coefficient for sediment eddy viscosity

K

von Krmn constant

w WI X

y

y0 Yl y+ z

a

c I

Is Ix yl 6

ISV C

Es

À li

1i

V

p

o -r

T0 Tc

Ii cJ

il)

lpp

porosity dynamic viscosity of fluid viscosity of sediment-fluid mixture kinematic viscosity, y = fluid density solid density standard deviation; normal stress shear stress boundary shear stress critical or threshold boundary shear stress; yield stress for granular medium shear stress at upper surface of density current angle of inclination Wadell sphericity maximum projection sphericity