COE CNG ee oe Gr Ve Sand Ripple Growth in an Oscillatory-Flow Water Tunnel by Karl E. B. Lofquist TECHNICAL PAPER NO. 78-5 | AUGUST 1978 Approved for public release distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING CB RESEARCH CENTER YSO Kingman Building Bae, a ied Fort Belvoir, Va. 22060 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 Contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. MBL/WHOI MN 0 0301 00899785 ec IM AON UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM T. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER TP 78-5 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED SAND RIPPLE GROWTH IN AN OSCILLATORY-FLOW Technical Paper WATER TUNNEL 6. PERFORMING ORG. REPORT NUMBER 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Karl E. B. Lofquist CERC Agreement No. 76-30 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK 3 AREA & WORK UNIT NUMBERS National Bureau of Standards National Engineering Laboratory Washington, DC 20234 D31193 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Coastal Engineering Research Center (CERRE-CP) 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 100 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 1Sa. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release, distribution unlimited. - DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Coastal processes Ripple profiles Equilibrium bed forms Sand motion initiation Oscillatory flow Three-dimensional bed forms Ripple growth Waves 20. ABSTRACT (Continue am reverse side ff meceasary and identify by block number) The development of sand ripples in an oscillatory-flow water tunnel was observed in 104 laboratory experiments approximating conditions at the seabed under steady progressive surface waves. The period, T, and amplitude, a, of the water motion were varied over wide ranges. Three quartz sands were used, with mean grain diameters, D = 0.55, 0.21, and 0.18 millimeter. In 24 experiments, with the bed initially leveled, T was reduced until ripples appeared, and their development to final equilibrium form was observed without further change in T. (Continued) DD anys 1473 EprTIon oF t wov 651s OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) The remaining 80 experiments investigated the response of previously established bed forms to changes in T or a or both. The ripple length, A, and height, n, were measured from photos, except when bed forms were three dimensional. Sand grains began to move on the flat bed at flow velocities generally com- patible with earlier observations. A Shields criterion for motion initiation was found in need of modification for use in oscillatory flow. Prior compres- sion of the bed had small effect. With the grains in motion, ripples were initiated by and would spread from slight irregularities on the bed or around its edge, or could be generated everywhere by a slight increase in velocity. Flow separation and vortex forma- tion occurred at an early stage. Rolling-grain ripples as described by Bagnold (1946) did not always form and seemed unnecessary to vortex ripple development. The initial length of the vortex ripples was small and remarkably independent Oe Ble During growth, the dimensionless ripple length and height, i/a and n/a, were expressible as functions of (D/a)n, where n is the number of periods since rip- ples first appeared on the flat bed. Ripple growth was approximately complete when (D/a)n = 1. Profiles of two-dimensional ripples in equilibrium with the flow were gener- ally as described by Mogridge and Kamphuis (1972). In particular, \/a and n/a were fairly constant over a wide range of intensity of flow. However, with end crests fixed, and a varying, a profile could be "'strained"' into various stable forms over a considerable range of A/a. Both two- and three-dimensional final bed forms were observed. Criteria for these forms were not clearly established, but appeared to depend on both the amplitude and intensity of flow, and the grain size. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report is published to improve understanding of the occurrence and form of ripples under water waves, since ripples can reduce wave height and affect sand transport. The report summarizes a laboratory study, including prototype offshore conditions, of sand ripple growth in sinusoidal flow. The results will enable the engineer to better predict and interpret equilibrium ripples in the offshore under both laboratory and field conditions. The work was carried out under the coastal pro- cesses program of the U.S. Army Coastal Engineering Research Center (CERC) . The report was prepared by Karl E. B. Lofquist, Physicist, Fluid Engineering Division, National Engineering Laboratory, National Bureau of Standards, Gaithersburg, Maryland, under CERC Agreement No. 76-30. Dr. R. J. Hallermeier was the CERC monitor for this agreement, under the general supervision of Dr. C. J. Galvin, Jr., Chief, Coastal Processes Branch, Research Division. Comments on this publication are invited. Published in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. JOHN H. COUSINS Colonel, Corps of Engineers Commander and Director II Til IV VI VII VIII CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) SYMBOLS AND DEFINITIONS . BACKGROUND. ‘ 1. Introduction . 2. Dimensionless Wessiebiles. 3. Previous Observations. EXPERIMENTAL PROGRAM AND PROCEDURES . 1. Nature and Scope of the Experiments. 2. Ranges of Materials and Variables. 3. Referral to Prototype Conditions . INITIATION OF GRAIN MOTION. 0 1. Determination of the Condition . 2. Effects of Surface Compression . Sy RESUMES s FoOmsthemmnree SandSeira nr. c 4. Applicability of a Shields Criterion : EARLY STAGES OF RIPPLE FORMATION. 1. Induced and Spontaneous Ripples. : 2. Observations with Coarse and Finer Sands 0 3. Rolling-Grain Ripples. 4. Initial Ripple Length. GROWTH AND VARIABILITY OF EQUILIBRIUM RIPPLES . 1. Two- and Three-Dimensional Forms . 2. Growth of Two-Dimensional Ripples. 3. Equilibrium Two-Dimensional ee nORhes. 4. End Effects. : 5. Variability and Hysteresis of Stabile Promos. CONDITIONS FOR TWO- AND THREE-DIMENSIONAL BED FORMS . 1. Approach . 2a Resultse 3. Results with Ime Ss (1957) ice Rees Data 5 INTERPRETATION OF RESULTS . Growth of Ripples. Bee 9 Size and Shape of Stable Ripples c Two- and Three-Dimensional Bed Forms . Disappearance of Ripples Occurrence of Ripples. DAUnBWNE SUMMARY . Initiation of Grain Motion and Ripple ‘Formation. APPENDIX A DATA FROM 104 EXPERIMENTS ON RIPPLE GROWTH AND EQUILIBRIUM FORMS . B PHOTOS OF INITIAL AND FINAL BED FORMS FOR THE 24 EXPERIMENTS STARTING FROM AN INITIALLY LEVELED BED. TABLES 1 Symbols used to indicate combinations of D and N. 2 Symbols showing degrees of two- and three-dimensionality. FIGURES 1 Definition sketches for ripple dimensions . 2 Trends of 9, as function of a/D, from various observers . 3 Design curves for equilibrium bed-form wavelength . 4 Design curves for equilibrium bed-form height . 5 d/D as function of 2a/D, from various observers . 6 Longitudinal section through spool and the adjacent end of the test section . : 7 Grain-size distributions for the three sands. 8 Photos of grains of the three sands . 9 Distribution of N and T for experiments 1 to 57 with the 0.55-millimeter sand . 10 Distribution of N and T for experiments 58 to 84 with the 0.18-millimeter sand . Bae Ay Or 11 Pistribution of N and T for experiments 85 to 104 with the 0.21-millimeter sand . RID are pesiaies 12 Observed 4, as a function of a/D and r (or D) CONTENTS LITERATURE CITED . Page 83 86 92 34 66 13 16 19 21 25 29 31 32 35 36 OH 41 13 14 15 16 17 18 19 20 21 BO 2S 24 Z5 26 Zu 28 29 30 31 32 CONTENTS FIGURES--Continued Comparison of ~ computed from the data of this study with Shields curve for steady flow and a modified curve of Carstens, Neilson, and Altinbilek (1969) Rolling-grain ripples and vortex ripples. Initial ripple length, independent of a. A three-dimensional phase during growth . Response of a bed form to a reduction in a. Response of a bed form to an increase in a. Growth of ripple height with time, n/a as function of (D/a)n. Growth of ripple length with time . Equilibrium A/D as function of 2a/D . Equilibrium A/a as function of 9. Equilibrium n/a as function of 9 and a/D. Response of a profile to varying strain . Final n/dA for strained profiles obtained by varying a. Compressed, normal, and extended profiles. Occurrence of two- and three-dimensional bed forms with the 0.18-millimeter sand . Occurrence of two- and three-dimensional bed forms with the 0.21-millimeter sand . Pairs of final two- and three-dimensional bed forms at large and small N. Occurrence of two- and three-dimensional ripples on the seabed . Analogy between (a) transition from laminar to turbulent flow in a rough pipe, and (b) transition from oscillatory-flow to steady-flow ripples . Limits to the occurrence of ripples on the seabed . Page 68 70 Wi 76 79 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 Square centimeters cubic inches 16. 39 cubic centimeters feet 30.48 centimeters 0.3048 meters Square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 square meters cubic yards 0. 7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars 1.0197 x 105° kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! 1T9 obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F -32). To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. SYMBOLS AND DEFINITIONS amplitude of the water motion, one half the orbital diameter (2a) average grain diameter, equivalent to Dsg a descriptive grain diameter, meaning that m percent of the sand, by weight, is composed of grains with diameters less than Da bottom friction factor, defined by equation (23) acceleration due to gravity depth of water height of surface wave (trough to crest) surface wave number, = 21/L surface wavelength the number of periods, or cycles, since ripples first appeared on a flat bed, or since the start of an experiment with a bed previously rippled the number of turns of a nut setting the stroke of the pistons and, hence, a measure of the amplitude of the water motion (a = 0.4576 N centimeters) a dimensionless measure of the grain size, defined by equation (2) period of the sinusoidal water motion T for the initiation of grain motion on a flat bed T for the initiation of ripples on a flat bed amplitude, or greatest magnitude, of the water velocity above the boundary layer U for the initiation of grain motion on a flat bed U for the initiation of ripples on a flat bed dimensionless parameter defined by equation (7) ripple height (see Fig. 1) SYMBOLS AND DEFINITIONS--Continued Yipple length (see Fig. 1) initial length of ripples newly formed on a flat bed natural or unstrained ripple length kinematic viscosity of water density of water density of sand grains bottom stress t for the initiation of grain motion on a flat bed dimensionless measure of the intensity of sand motion, defined by equation (1) o for the initiation of grain motion on a flat bed ¢ for the initiation of ripples on a flat bed >| for the onset of sheet flow Shields parameter, defined by equation (21) angular frequency, = 21/T SAND RIPPLE GROWTH IN AN OSCILLATORY-FLOW WATER TUNNEL by Karl E.B. Lofquist I. BACKGROUND 1 Introduction. Oscillatory surface waves commonly produce ripples on a sandy seabed. These ripples, on the one hand, increase the roughness of the bed and so increase the bottom stress and the rate of energy loss from the waves which produced them. On the other hand, they enhance the agitation of the sand and so are an essential part of the mechanism of sand transport. When the oscillatory water motion over the bed is able to move sand grains along the bottom or to lift them above it, if only during a part of the cycle, a superimposed steady current, even though weak, is able to move sand consistently in one direction. Ripples enhance the movement and lifting of sand grains, particularly near their crests where water velocities are amplified and where the sharp curva- ture of the profile serves to separate the flow and the moving grains from the bottom. The involvement of these effects in coastal engineering has en- hanced interest in oscillatory-flow ripples, which have claimed much study. Nevertheless, these ripples remain poorly understood. This is in large part because observations remain inadequate to define the roles of the many factors involved. Also, the import of observations is some- times obscured by diversities in technique and, less often, by apparent discrepancies. This study describes a series of laboratory experiments. Its purpose is to add to available observations with special attention to ripple initiation and growth. Observations and discussions of ripples usually concern two- dimensional bed forms and one or another of several stages in ripple growth from an initially level bed. These stages are: (a) Initiation of grain motion; (b) possible regimes of rolling grains or "rolling-grain" ripples; (c) emergence of "vortex" ripples, spontaneously or from rolling-grain ripples; (d) subsequent growth of these vortex ripples; (e) attainment of stable equilibrium form; and (f) erosion and ultimate disappearance of bed forms at high water velocities. This experimental study deals with all these stages in varying degree, as have the previous investigations which are discussed in Section I,3. Sketches of regular and irregular ripple profiles with definitions of their dimensions are shown in Figure 1. Die Dimensionless Variables. Interpretation of ripple observations is made easier by the use of dimensionless variables. This study adopts, with slight modification, the system advanced by Yalin and Russell (1962). The four independent dimensionless variables are: 2 ee DN ee Sr ase ae 2 Cota) D (1) i/ (<= —1)p3 T= « (2) y2 where 2a = orbital diameter of the fluid motion D = average grain diameter Pas Pheus densities of sand and fluid g = acceleration due to gravity U = maximum velocity of the fluid v = kinematic viscosity of the fluid /}—— 2a = orbital diameter of water motion es (a) (b) Figure 1. Definition sketches for ripple dimensions, showing the measurement of n for regular (a) and irregular (b) profiles. It is convenient to express the effects of viscosity in terms of r rather than by the more usual grain-size Reynolds number, UD/v, equal to r¢!/2, because r depends only on the materials used and is typically a constant during a series of experiments. When the materials involved are quartz sand and water and p

@- a (sand motion) ) > = 6.6, 58 =) (ripple formation) (4) and, for example, the dependent equilibrium ripple length i can be expressed as KM _ a a Ps D =~ pld> pe ae. . (5) Thus, the criteria for the initiation of sand motion, and of ripple formation, are described by dimensionless plots without distinguishing whether the boundary layer, or stress, is laminar or turbulent. Such distinction has been attempted by Komar and Miller (1973) and others. The degree of turbulence can be defined by the character of the stress coefficient which is itself some function of the above variables. Be Previous Observations. a. Facilities. Results of previous investigations may best be introduced by a brief evaluation of facilities used in the investiga- tions most often cited. Bagnold (1946) and Manohar (1955) used sediment-filled trays oscillating in still water. This device conveniently provided wide ranges of period and amplitude, but subjected 14 the grains to unnatural (though possibly negligible) accelerations. Stationary beds are provided by wave tanks and oscillatory-flow water tunnels. Inman and Bowen (1962), Kennedy and Falcon (1965) and Yalin and Russell (1962) used wave tanks. Although the mechanics of sand movement by waves in tanks are that of the prototype, limitations in tank size and wavelength keep wave periods rather small, typically less than 3 seconds. Oscillatory-flow water tunnels may or may not be limited in period. Carstens, Neilson, and Aitinbilek (1969) used a tunnel with a rather wide cross section well adapted to observe three- dimensional bed forms but which was restricted to a single period of oscillation (3.56 seconds). Mogridge and Kamphuis (1972) and Rance and Warren (1968) had the use of large tunnels without restriction on period. The tunnel of Chan, Baird, and Round (1972) was limited in its range of period and by a circular cross section only 5 centimeters in diameter, but was suitable for use with fluids other than water. Inman (1957) and, later, Dingler (1975) observed ripples on the seabed itself. The seabed lacks all the usual laboratory restrictions of size and scale, but it is out of the observer's control so that the cause of an effect may be obscured. Both Inman (1957) and Dingler (1975) discussed the problem of defining an amplitude and a frequency for the bottom water motion for field conditions where the wave spectra have many components. b. Initiation of Grain Motion. The first stage of ripple develop- ment, the initiation of grain motion, has been the most studied. Silvester (1974) provides references to 13 separate criteria, to which may be added others of Komar and Miller (1973; 1975), Chan, Baird, and Round (1972), Sternberg and Larsen (1975), and Dingler (1975). Of the assembled criteria, 14 can be put in the dimensionless form, o = const. 2% k (6) This particular form of equation (3) is the usual result of fitting data, over a limited range, with a straight line on a log-log plot. Some of the criteria apply specifically to laminar or turbulent boundary layers. Criteria which can be defined by the data of Bagnold (1946) and Manohar (1955) have been discussed by Komar and Miller (1973). However, taken altogether, the available data do not clearly define the function $.(a/D, r, 0 107; and from Chan, Baird, and Round (1972) who, using a small pipe but with various viscous liquids, attained values of r < 0.2. The various trends reflect a considerable scatter in the data, on the order of + 40 percent. Some of this scatter may derive from the neglected effects of grain-size distribution and grain shape which, as Collins and Chesnutt (1976) describe, can be significant in the evolution of beach profiles or from the degree of compaction of the sand surface. However, most of the scatter can be attributed to the subjective element in determining the condition of incipient grain motion. The definition and determination of Uc. and o, are discussed in Section III,1. Generally, the data represented in Figure 2 show a weak correlation of larger values of ¢c¢ with smaller values of r. For convenient reference, a single composite curve, for all values of r, has been drawn through the trends shown in the figure. For small a/D, this curve becomes parallel to the trend of Rance and Warren (1968), with $, proportional to (a/D)!/3; for large a/D, the curve makes $. proportional to (a/D)2/3, a behavior suggested by the known stress on a smooth bed combined with a Shields criterion, as defined in Section III,4. . c. Early Stages of Ripple Development. Observations of ripple development following the initiation of sand motion as reported by various investigators are summarized below. Bagnold (1946) identified a "rolling-grain" type of ripple stable over the range U. < U < 2U, with ripple length and height, for constant a, increasing with w. At U®2Ue these ripples rather suddenly change character and become "vortex-type" ripples, with vortices developing behind the ripple crests and over the troughs. The vortex ripple length is independent of w and is proportional to a, for small a, and constant for larger a. Bagnold states that these ripples can form ''from any sufficiently large surface feature" at values of U even below U,. He describes artificial generation of ripples for U < U, (also described in more detail by Carstens, Neilson, and Altinbilek, 1969). Manohar (1955) describes stable regimes of rolling-grain motion and "first-stage" Yipples very similar to the rolling-grain ripples of Bagnold. He also describes ''second-stage'' ripples which appear to be the vortex ripples of Bagnold. Throughout the process of ripple formation, Manohar finds, for given a, stable ripples at each value of w, and ripple length increasing with w. He provides data for the "initiation of ripples" which occurs, on the average, at a value of U which exceeds the values for "initial" and "general" motion of "sediment in the turbulent boundary layer'' by 24 and 13 percent. Presumably, this refers to the initiation of second-stage or vortex ripples. Kennedy and Falcon (1965) provide no data on the inception of either sand motion or ripple forma- tion, but observe that, when there is ''general motion of the upper layers" of the bed particles, ripples appear simultaneously and ''once they grow to a certain height, vortices are shed from the ripple crests." They state that, in their experiments, the rolling-grain ripples described by Bagnold (1946) did not appear with quartz sand but did appear with plastic (opalon) beads. They also report that ''the equilibrium ripple wavelength was observed to form at an early stage of ripple growth for all sediment materials used" and that the ripple length increased with both a and w. Carstens, Neilson, and Altinbilek (1969) describe an (apparently) unstable progression from rolling grains through a stage of rolling-grain ripples and into the stage of vortex ripples, which attain an equilibrium length, approxi- mately proportional to a. They find a "small lag" between incipient motion and the spontaneous formation of ripples which, by an average of their three observations, occurs around U = 1.16 Uc. d. Equilibrium Two-Dimensional Ripples. The preceding descriptions include references to the stable forms which ripple profiles attain. The primary source of information on two-dimensional equilibrium ripple forms is the study of Mogridge and Kamphuis (1972), obtained with a large oscillatory-flow water tunnel and using three types of plastic grains and a single quartz sand (with D = 0.36 millimeters). Observa- tions of the equilibrium ripple length and height are summarized by curves in Figures 17 and 18 of their report. These figures give i/D and n/D as functions of 2a/D for selected values of the parameter, i D oY fD ND rc 7 ; Be = Or re (2a) ce Figure 17 of Mogridge and Kamphuis (1972) is reproduced in Figure 3 with the addition of lines of constant drawn according to equation (7). The figure shows that, over wide ranges of a/D and 6, lines of constant ¢ are approximately straight with a slope of plus one. Thus, along each such line, A/a is approximately constant and independent of a/D. That is, r ~ a @ fd) - (8) As @ increases, \/a ultimately decreases while its dependence ppen a/D increases gradually. In particular, 102 < =< 5x103 (9) o|> 2 wl ) 2 Equation (9), remarkably, states that \/a is nearly constant and inde- pendent of U or T over a wide range of 4, which (as this study confirms) o|> Figure 3. Design curves for equilibrium bed-form wavelength. A/D is given as function of 2a/D and X, by Mogridge and Kamphuis (1972), and curves of constant $ have been added. 9 and X, are defined by equations (1) and (7). extends to high rates of scour and high degrees of sand agitation. Similarly, Figure 18 of Mogridge and Kamphuis (1972) is reproduced in Figure 4 with lines of constant $ added. Again, for small values of ¢, n/a is approximately constant, but, as 4 increases, n/a decreases and its dependence upon a/D, increases, both relatively rapidly. As @ increases, n/a decreases more rapidly than does i/a, so that the ripple steepness, n/d decreases. In Figure 5, A/D as function of 2a/D, from several studies, is compared with the envelope curve of Mogridge and Kamphuis (1972). Kw 2(5*) | (10) Examination of the sources of the plotted data reveals that the effect of ¢$, or of Xp, is seldom such as to displace the points significantly from the envelope, so that displacements are to be interpreted as disa- greement or as scatter. The studies included in Figure 5 are: (a) Bagnold (1946), using an oscillating tray in still water and grains of quartz, coal, and steel. (b) Carstens, Neilson, and Altinbilek (1969), using an oscillatory-flow water tunnel with quartz sand, flint shot, and glass beads. (c) Yalin and Russell (1962), using a wave tank and grains of quartz, coal, and Lucite. (d) Kennedy and Falcon (1965), using a wave tank and grains of quartz (two sizes) and two plastics. (e) Inman and Bowen (1962), using a wave tank and a single quartz sand; also Inman (1957), observing in the field. (f) Dingler (1975), using a wave tank and quartz sand (two sizes) and observing in the field. Dingler (1975) discusses most of these studies and presents plots similar to those in Figure 5. In general, for 2a/D < 2x103, the data in Figure 5 agree with the curves of Mogridge and Kamphuis (1972) as reduced to equation (9) or (10). An exception is the leveling off of the curves of Bagnold (1946) in Figure 5(a). Since laboratory observations in Figure 5 for 2a/D < 2x103 include grains of quartz over a wide range of sizes, and grains of other materials, they extend as well as confirm the results of Mogridge and Kamphuis (1972). 20 AMRUTG UAT ee eT rime o|s Figure 4. Design curves for equilibrium bed-form height. n/D is given as function of 2a/D and X, by Mogridge and Kamphuis (1972), and curves of constant ¢ have been added. 4 and X, are defined by equations (1) and (7). 2| Ol“ Oo ol Figure 5. Bagnold (1946) ye Corstens, Neilson, and We Altinbilek (1969) io? | e in. 4 T T ST T (a) (b) 10) 1@) 1 1 1 107 ees (Cc) (d) Yolin ond Russell (1962) Kennedy and Falcon (1965) lof | ca ee om r Acad 4 107 ae ee T (e) (f) + Inman and Bowen (1962) wa Dingler (1975) Ve + Inmen (1957) ORE 3 | io? oes F + fob oie A 250 - field t f Y 0% tf ° 10% 4 10 10° 10° 10* 10 10? 10° io* 2a 2a D D 4/D as function of 2a/D, from various observers. The straight line in each plot is given by equation (10). ‘However, for 2a/D > 2x10%, the data in Figure 5 show a tendency for \/D and A/a to fall below values according to equations (10) and (9). This tendency appears in the laboratory observations in Figure 5(b and d) and, more strongly, in the field observations in Figure S(e and f). Circumstances, especially in the field, appear to favor a positive correlation between the "independent" variables 2a/D and 9, so that their effects are not readily separated. This problem is dis- cussed further in Section VII,6. The data falling below the envelope curve of Mogridge and Kamphuis (1972) in Figure 5 include large values of $4, between 50 and 200, which can account for some of this deviation. However, the reductions in \/D thus accounted for are a rather small part of those actually found, and substantial reductions are found also with small values of 94. Thus, the data, with their scatter, reveal no clear effect of upon A/D. Dingler's (1975) data, unlike those of Inman (1957), form two separate clusters, despite fairly continuous distributions of 2a/D and ¢. Small values of A/a appear to be characteristic of small sand grains, and values of A/a < 1/2 are most often associated with values of D < 0.2 millimeters. This would suggest an effect of grain size, or of r, mot included in the curves of Mogridge and Kamphuis (1972). In their experiments, r remained within the rather narrow range 28 < r < 43, so that its possible effects were hardly explored. Nielsen (1977) has suggested that A/a should be a function of U/w, where w is the fall velocity of the sand grains, and has plotted data from various sources in this form. His curve drawn through the data shows an accelerating decline in \/a with increasing U/w similar to that with in equation (8). U/w can be written as ot 2£(r), where the function f(r) depends on the variation of the drag coefficient of the grains with Reynolds number. Over the range of practical interest, f(r) decreases with r almost as rapidly as l/r. Then, a reduction in r, as well as an increase in 9, increases U/w and so, by Nielsen's curve, reduces A/a. Such a behavior seems appropriate for a modification of the curves of Mogridge and Kamphuis (1972) with (U/w)2 replacing 9. A rather large degree of scatter in the data of Nielsen's plot obscures to what extent separate effects of » and r combine into the single effect of U/w. However, Nielsen (1977) has identified U/w as a variable likely to be useful in analyses of ripple forms. Several studies (not included in Fig. 5) report behavior of A quite different from the results of Mogridge and Kamphuis (1972). Manohar (1955), using an oscillating tray in still water, found i/D to be almost independent of a and to be approximately proportional to U. Guided by the process of vortex shedding by cylinders, Homma and Horikawa (1962) cite the field data of Inman (1957) and their observations and those by Scott (1954) in wave tanks to show that A/a is primarily a function of the Reynolds number Ua/v. Horikawa and Watanabe (1967) obtained a similar result when sand was replaced with plastic pellets. 23 However, the data of Scott (1954), within scatter, are fairly con- sistent with equation (10), and the data of Inman (1957) in Figure 5(f) show increased scatter when plotted against Ua/v. On the basis of theory and some observations, Shulyak (1969) found that he CU Wyatt (11) where C and U, are functions of grain and fluid properties. The presence of U, makes this expression incompatible with equations (9) and (10). e. Disappearance of Ripples. Observers agree that, as a, or U, or both increase, bed forms reach a maximum height and then decline and eventually disappear in a slurry, a condition Dingler (1975) calls "sheet flow." At the onset of this condition, » = $5. Kennedy and Falcon (1965), using Inman's (1957) data, find this disappearance at a/D = 8,000 (UT/D = 50,000), while Carstens, Neilson, and Altinbilek (1969) find a/D = 1,700. Earlier, Carstens (1966) described the growth and decline of bed forms in terms of gi /2 rather than a/D, with maximum size at o!/2 = 6.5 and disappearance at o!/2 = $9,!/2 = 13.0. In any case, the tunnel in Carstens' studies was restricted to a single period (3.56 seconds), so possible separate effects of @ and a/D could not easily be distinguished. Mogridge and Kamphuis (1972) observe that bed forms decrease and disappear with increasing a/D, but seem to imply that T (with Xo) is here held constant, so that U increases with a. In Figure 4, as $ increases n/D decreases and can reasonably be extrapolated to zero. Manohar (1955) (with Figure 23 in his report) provides a crite- rion for the disappearance of bed forms which can be written (el eS = 20.9) For each grain material (value of r), Manohar (1955) obtained U, by averaging all observations, each at a given value of a, and thereby obscured any dependence of ¢, on a/D. With all observations plotted, the data (in Manohar's Table 9) show We clearly increasing with a for eight grain materials, with scatter obscuring the trend in the single remaining case of the lightest plastic. For glass beads and quartz sand (2.5 < 9

a/D. Other stages of ripple growth can be defined in similar terms. Such criteria, which make the time of development proportional to {a/D)T, appear to be in conflict with Inman's (1957) comment that ripples in fine sand are more rapidly modified than ripples in coarse sand. It is not clear under what conditions or restrictions, if any, this comparison applies. In any case, U need not approximate Uc as in Figures 19 and 20. Only three experiments with the finer sands are included in these figures and some further dependence upon grain size (r) may well exist. Along the asymptotes in Figures 19 and 26, the data show considerable scatter which is largely due to erid effects (discussed in Sec. V,4). Sudden and substantial increases in i and n result when a shrinking crest finally ceases to be counted. These increases result from averaging over the limited number of ripples in the test section and do not imply actual abrupt changes in the profile. Si Equilibrium Two-Dimensional Ripple Profiles. Photos of the final equilibrium profiles for all the experiments started from a leveled bed are contained in Appendix B. The two- dimensional profiles differ widely in scale, but are rather similar in shape, with variations of types discussed below. The photos reveal a variety of configurations at the ends of the profiles. The scour some- times appearing at the curved ramp acting as a persistent trough, may serve to ''anchor'' the end of the profile. Final equilibrium values of X/D are plotted against 2a/D in Figure 21. Included in the data are all the experiments with the 0.55- millimeter sand, and 33 experiments with the 0.18- and 0.21-millimeter sands which were sufficiently two dimensional to permit analysis. In this form, the data can be compared directly with the previous observa- tions in Figure 5, by their relation to equation (10) which is the straight line common to both Figures 5 and 21. The data of this study (Fig. 21) follow this line reasonably well, though their trend shows A/D to increase, nearly, with the three-fourths power of 2a/D, rather than with the first power. Deviations of the data from equation (10) are not accounted for by the regular and relatively minor effect of $6. However, the deviations are within the range of inherent scatter and end effects discussed in Section V,4. Also, values of A/D falling below the line for the four highest values of 2a/D may be the beginning of the trend for }/a to decline with 2a/D, which was found in several of the plots in Figure 5, and discussed in Section I,3,d. Here, as 55 5xlO 2a D Figure 21. Equilibrium A/D as function of 2a/D. The straight line is given by equation (10). 56 in those cases, sands with small grain sizes are involved. Further, these four experiments were among those with profiles showing some three dimensionality, which is characterized by smaller bed forms, and in two of them (experiments 64 and 82 with the 0.18-millimeter sand) an occasional recurrence of three-dimensional bed forms may have prevented attainment of two-dimensional equilibrium. With these considerations and the previous observations (Fig. 5), the data in this study are regarded as agreeing with equation (10) reasonably well, rather than as indicating a three-fourths power law. To observe the effects of 6, which were not revealed in Figure 21, the same data have been replotted in Figure 22 to give i/a as function of . Flags have been added to the plotted symbols to give values of N (or a/D) as shown in Table 1. The curve in Figure 22 is another replot of Figure 17 in Mogridge and Kamphuis (1972). This curve indicates that in the experiments of this study the effect of $ upon i/a remains slight but still dominates that of a/D which equations (8) and (9) show to be negligible. Final equilibrium values of n/a are plotted against » in Figure 23. Included experiments (with values of and symbols) are the same as in Figure 22. The curves for constant values of a/D in Figure 23 are a replot of Figure 18 in Mogridge and Kamphuis (1972). In this case, over the range of the data, the effects of varying a/D are noticeable, and separate curves have been drawn. However, the curves form a rather narrow band, and do not serve to reduce the scatter. The data do not follow the curves as closely as in Figure 22, with points for the coarse sand falling above and points for the finer sands falling below. These moderately separate trends indicate a noticeable dependence on r not contained in Figure 18 of Mogridge and Kamphuis (1972). 4. End Effects. Figures 19 to 23 show a considerable degree of scatter, attributable to variations in the effects of the ends of the channel on the average ripple length. Assuming that the particular geometry of the ends some- how tends to hold crests in certain fixed positions near the ends (as does the local scour before a ramp), two such crests, one at either end of the channel, will be ''fixed" a distance S apart. To be stable, the profile between must have an average ripple length A = S/m where m is some integer. There is usually no integer which makes i equal to Ay, the 'natural'' ripple length which, for the given a and T, would form on an unbounded bed. If A, > the profile will tend to press the end crests outward or, if Ay < A, to pull them inward, increasing or decreas- ing S, and in either case acting against the end effect tending to hold the crests in place. Thus, the resulting displacement of the crests is limited and leaves i still unequal to ’,. Preferably then, the ends should be indifferent to the position of the end crests and, if space had allowed, long gently sloping ramps might have been preferable to the 57 a|» Mogridge and Kamphuis (1972) a 250 Sy < 1500 PLETE. 22. Equilibrium A/a as function of 9. ? The data are in general agreement with Mogridge and Kamphuis (1972). Symbols are defined in Table 1. 58 als Mogridge and Kemphuis (1972) Figure 23. Equilibrium n/a as function of ¢ and a/D. The data indicate an apparent effect of grain size not found in Mogridge and Kamphuis (1972). Symbols are defined am WAS Ike 59 short curved ones actually used (Fig. 6). In present circumstances, not only the ramps with scoured troughs and crestlike summits, but the screens in the spools act to position and hold crests, since the sand which cannot pass through a screen tends to accumulate a short distance in front of the screen, in the form of a crest (Fig. 6). Such crests are liable to move, and cases of cyclic instabilities of sand crests within the spools, with an otherwise stable profile, were observed. Finally, it seems that a stable profile cannot end abruptly, but requires the accumulation of a certain amount of sand in the spools to provide a gradual transition to the unrippled region beyond. This accumulation usually occurs by migration of a ripple crest into the spool where the crest is rearranged into separate peaks on the bare floor of the spool. The migration of the ripple crest into the spool may extend the profile and make A > Ay. This effect was tested in experiment 57 by interrupting the experiment before the ripples reached equilibrium and adding a quantity of sand, by hand, into each spool. The subsequent equilibrium i was found to be reduced by 20 percent from that in experiment 29 with the same a and T. In experiments 30 to 50, pieces of brass bent into a crest shape, 2.8 centimeters high and with a vertex angle of 120°, were mounted in the spools at various posi- tions between the screens and the ends of the sand bed. The record for experiments 29 to 36 in Table A-1 shows that 2 was reduced by moving the brass crests forward in the spools, while the redistribution of sand in experiment 31 had a relatively small effect. Thus, the brass erests provade) a rellatively, definite), atvextremes endvertect 3a Wax the brass crests removed, the end effects remain unclear in nature and detail but appear to have an order of magnitude quite able to account for the scatter in Figures 20, 21, and 22. Sy Variability and Hysteresis of Stable Profiles. Although the mechanism of the end effect could not be analyzed, it was possible to measure its limits. A series of experiments (37 to 47) with the 0.55-millimeter sand was performed to determine over what range of A/a a profile could remain stable and how the profile shape varied with \/a. The brass crests were mounted atop the ramps 254 centi- meters apart with seven sand crests between. Since the brass crests fixed the ends of the profile, the average ripple length, A, remained constant in this series (equal to 254/8 = 31.8 centimeters) as long as the number of sand crests remained unchanged. As in equation (9), A, is approximately proportional to a, with /d, proportional to A/a. Each experiment of this series had a different value of a (and hence A/a) and T was adjusted to keep U approximately constant at 34.5 centimeters per second ~U,.. The variable n was observed. Photos of the center third of the final profiles for these experiments are shown in Figure 24. Figure 25 is a plot of final n/d as function of i/a including the numbers of the consecutive experiments. The series begins at experiment 37 with a low }4/a; that is, a compressed profile. In experiment 38, with A/a further reduced to about 1.1, signs of instability appeared (note the incipient shrinking of the center crest in Fig. 24) and the experiment was quickly stopped. Previous experiments had shown that, if the 60 Figure 24. Response of a profile to varying strain (the mid-third of test section in experiments 37 to 47). Numbers by each photo give (in descending order) the experiment number; N; the number of cycles since the beginning of the experiment (n); and (D/a)n. (D = 0.55 millimeters.) 6| 380 45 IS 013 "compressed" "extended" stable | | | \ | | | | \ | | I | | | | | | | | ! | range | | a ee eee fl Q|’ Figure 25. Final n/A for strained profiles obtained by varying a. For the numbered experiments, 2 and U are held constant. (D = 0.55 millimeters.) 62 experiment were continued, one crest of the compressed profile would shrink and disappear (and thereby increase A). In following experi- ments, A/a was successively increased, and the observed n/\ successively decreased. The troughs became longer and shallower, and each eventually developed a gentle "rise", characteristic of an extended profile. In experiment 44 one of the trough rises began to grow. The experiment was stopped but apparently not soon enough; in the following experiment 45, even with A/a reduced to 4/3, this growth, though slower, continued. Not until experiment 46, with X/a in the compressed range, did the new crest shrink and disappear. The number of sand crests was still counted as seven and A unchanged in determining i/a and n/d for experiments 45 and 46. The appearance and disappearance of this incipient crest (Fig. 24) give the curves in Figure 25 a hysteresis which would not have occurred had \/a been kept below some value between that for experiments 43 and 44 and the new crest not allowed to start. Following experiment 46, A/a was again increased, and the point for experiment 47 falls on the former descending curve. On the basis of these observations, the stable ranges of i/a and n/d are taken to be, approximately, Lol < Wa < 16 (24) (stable) O22 > V\ S O12 (25) The middle of the range of X}/a in equation (24) is near 4/3, so that An can be identified as the ripple length predicted by equation (9). In Figure 25 the profile has been described as ''compressed" or "extended" as \/a is less than or greater than 4/3. The points from this series of experiments with their variations in \/a and n/a, are identified by the diamond symbol in Figures 22 and 23. It is seen that their ranges are about the same as the ranges of the scatter. Therefore, the scatter is probably associated with various degrees of profile strain. Characteristics of ''strained" profiles are shown in Figure 26. As in experiments 37 to 47, i is constant and, from top to bottom, d/a increases as a decreases. Based on the information summarized in Figures 25 and 26, the following implications for equilibrium ripple shape are drawn. When the end crests of a two-dimensional profile are fixed, the conditions of strain imply that changes in i responding to variations in a must show hysteresis. As a increases, such a profile becomes compressed and, as crests disappear, i increases in steps such that i > l.la (eq. 24). As a decreases, the profile becomes extended and as new erestsuappeanw ) (decreasessingstepsisuch that 1 <) 26a. (thus, A depends, in part, on whether a _ has been increasing or decreasing and must show hysteresis as a is varied from and returned to its original value, provided that the variation in a was large enough to eliminate an old crest or create a new one. The photos in Figures 17 and 18 show a 63 Figure 26. “compressed” = —— "extended" Compressed, normal, and extended profiles. From top to bottom, A/a increases as a is decreased. The dashed curves are the "normal' profiles redrawn for comparison. 64 hysteresis in the differences between the profiles at the start of experi- ment 48 (following experiment 47) where X = 31.8 centimeters, and at the end of experiment 49, where A = 25.4 centimeters. As the record in Table A-1 shows, both profiles are in equilibrium with the same combina- tion of a and T, but in experiment 48, a was half the value of experiment 47, and in experiment 49 a was returned to the original value of experiment 47. The first profile in Figure 17 (resulting from experiment 47) is unstrained with seven crests on the bed. The last profile in Figure 18 (resulting from experiments 48 and 49 in sequence) is compressed with nine crests. For the same variation in a, the hysteresis, or net change in A, would have been more had the first profile of Figure 17 been extended with only six sand crests on the bed. In the few possible cases where hysteresis might have occurred with the brass end crests removed, its observation is obscured by scatter in the data in Table A-1. VI. CONDITIONS FOR TWO- AND THREE-DIMENSIONAL BED FORMS les Approach. As described in Section V,l1, with the 0.55-millimeter sand the final bed forms were always two dimensional, but with the 0.18- and 0.21-millimeter sands the final bed forms were either two or three dimensional or even some combination of both. Although experiments with the finer sands yielded data for the preceding descriptions of two- and quasi two-dimensional profiles, they were also intended to identify the conditions for two- and three-dimensional final bed forms. To this end, the experiments were located again on plots of N and T, as in Figures 10 and 11, with symbols denoting the two- or three- dimensional character of the final bed forms. As before, N isa measure of amplitude a (a = 0.4576 N centimeters). The symbols and definitions are listed in Table 2. Although the definitions are sub- jective, it is expected that, after the symbols are plotted, a curve can be drawn separating two-dimensional from three-dimensional bed forms. 2. Results. Symbols showing the character of the final bed forms of the experi- ments with the 0.18- and 0.21-millimeter sands have been plotted in Figures 27 and 28 at the points (T,N) which define the experiments. These points are the same as in Figures 10 and 11 from which the number of each experiment can easily be obtained. In Figure 27, regions of predominantly two- and three-dimensional final bed forms can be separated by the straight line U = 25.2 centi- meters per second, with two- and three-dimensional forms on the sides of 65 Table 2. Symbols showing degrees of two and three dimensionality. Final bed form S ymbol 2D Trend to 2D Mosfly 2D 2:D;3D Mostly 3D Trend to 3D 3D 66 T (s) Figure 27. Occurrence of two- and three-dimensional bed forms with the 0.18-millimeter sand. Three-dimensional forms occur above the solid line, given by > = 21.3 or U = 25.2 centimeters per second. Symbols are defined in Table 2. 67 / / / / ee a”) 6 ie o000 0g / / / / / T (s) Figure 28. Occurrence of two- and three-dimensional bed forms with the 0.21-millimeter sand. Three- dimensional forms occur above the solid line. Symbols are defined in Table 2. 68 smaller and larger U. With D = 0.18 millimeters, this line can also be defined as a/D = (222 sec-1)T, or as ¢ = 21.3. The first defini- tion confirms the observation of Carstens, Neilson, and Altinbilek (1969) that bed forms are two or three dimensional as a/D is less than or greater than 775, when T = 3.56 seconds (the period in their experi- ments), but conflicts with their criterion as stated in terms of a/D alone. The second definition conforms generally, though only roughly, with Carsten's (1966) earlier criterion that two- and three-dimensional forms occur as $2 is less than or greater than approximately 6.5. In Figure 28, with the 0.21-millimeter sand, a line of constant 9 again appears to separate two- and three-dimensional regions for moderate and large values of a, but it cannot be extended to small values of a, where a two-dimensional character seems dominant at all values of T. In this case, for small values of a/D, the criterion of Carstens, Neilson, and Altinbilek (1969) seems to apply. A curve separating two- and three-dimensional regions has been attempted in Figure 28, but its contorted character warns of uncertainty. Why, at small values of a, two- dimensional bed forms should occur at small values of T with the 0.21-millimeter sand and not with the 0.18- millimeter sand is difficult to say. It can only be suggested that the relatively broad grain-size distribution of the 0.21-millimeter sand (Fig. 7) makes the dependence of bed form upon grain size less precise, and that the larger grains act to place the 0.21-millimeter sand in an unstable transition zone between the 0.18- and 0.55-millimeter sands. For the 0.55-millimeter sand, all final bed forms were two dimensional. Photos of final bed forms in four experiments with the 0.18- millimeter sand are shown in Figure 29. The experiments have been selected in two widely separated pairs at large and small values of a (at N = 100,25). Each pair straddles the line of constant @$ which separates the regions of two- and three-dimensional bed forms in Figure 27. The photos show that in each case the character of the bed form confirms the choice of the symbol plotted in Figure 27 and that the line of separation splits each pair. So Results with Inman's (1957) Sea Floor Data. An attempt was made to establish conditions for the occurrence of two- and three-dimensional ripples on the sea floor based on the observations and definitions of Inman (1957). As mentioned earlier, he defined the ripple pattern according to whether the length of the ripple crests was greater than 8) (long-crested), between 3A and 8), or less than 3\ (short-crested). Symbols for these patterns have been plotted by values of a/D and ¢ which, for 95 observations, could be obtained from the data in Inman's tables (see Fig. 30). In general, the three patterns are found to be mixed together over the (a/D,) field of 69 (*stozoUTJUSD N 9LGp°0 = B SSLOJOWTTTIW gtT°O = d) ‘N [Tews pue o81e[ }e SULOF Poq [BUOTSUSWTP-s91y. pue -oM} [eUTF FO sateg "6Z eANndTy 70 5 x10" crest symbol short (e) o e intermediate e 1.0 medium long 1 nn nnn 4 4 oO|a Figure 30. Occurrence of two- and three-dimensional ripples on the seabed (data from Inman, 1957). 7| the data, which were obtained mostly in exposed coastal waters. However, there appears to be some tendency for longer crested ripples to associ- ate with larger values of ¢ and a/D. Of the 5 observations for ¢ < 4, none is long crested; of the 27 observations for ¢ > 46, none is short crested, and all of the 4 observations for » > 110 are long crested. Also five observations for concitions of limited fetch, with T < 3.5 seconds, form a separate group displaced, at given 4, toward smaller a/D. These are all long-crested ripples. Inman (1957) found a strong tendency for long-crested ripples to occur nearshore in shallow water, which would imply an association with larger values of ». However, the effect of depth appears so strong as to suggest some factor beside 94, such as the increasingly two-dimensional character of surface waves as they advance over a shoaling bottom. VII. INTERPRETATION OF RESULTS We Initiation of Grain Motion and Ripple Formation. The observations of the initiation of grain motion on a flat bed in these experiments have provided values of ¢, as functions of a/D and r, as in equation (3). These data, and a separate curve for each value of D, or of r, are shown in Figure 12. It is seen that to suppress the effects of r and to approximate all the data with a single straight line on the logarithmic plot would introduce new scatter. Approximations of this sort certainly account for a part of the large disparity between trends in the data compiled by different observers, as indicated in_Figure 2. However, the major part remains attributable to the subjective element usually involved in determining ¢,., as discussed in Section III,1. This element can account for the present values of $¢ plotting somewhat above the composite curve in Figures 2 and 12. The effects of surface compression, investigated in two experiments, do not appear very significant. The applicability of a Shields criterion was tested with the obser- vations in this study. Application of the criterion involves a computa- tion of the oscillatory bottom stress which remains somewhat uncertain. Although the successful use of a Shields criterion does provide insight into the mechanics of incipient motion, the prediction of incipient motion can be approached directly, and perhaps more easily, by the empirical determination of ¢, as a function of a/D and r. Carstens (1966) has stated that the concept of incipient grain motion on a flat bed is of doubtful relevance in the formation of ripples, since the seabed is seldom flat, and he would suggest a study of the minimum conditions for the stimulation of ripples by foreign objects and it may be added here, by relict bed forms. However, the innumerability of foreign objects able to stimulate ripples, including 72 their dependence on orientation and exposure to the flow, makes such a study infeasible. Also, a distinction must be made between setting sand in motion at points on a relict bed form and establishing a new bed form. A new oscillatory flow may at first move sand on sharp relict ripple crests, but such flow may merely round the crests until the sand ceases to move. In predicting stable bed forms, it is more pertinent to deter- mine the minimum conditions of flow required to sustain them, assuming that any stimuli which may be needed to initiate their development will always be present. Minimum conditions to sustain a rippled bed can reasonably be sought in terms of a minimum value of 9. Lofquist (1975) found that an oscillatory flow, combined with an effect of permeability, became unable to sustain ripple migration at a value of around one- fourth ¢,.. Carstens, Neilson, and Altinbilek (1969), for a/D = 139 (in their experiment 62), identified a minimum $ = 2.15 needed to continue the growth and spread of small ripples initiated by a foreign object. A reduced ¢$ could possibly have maintained the ripples in equilibrium after they had attained full size. It was found that the conditions required to initiate grain motion on a leveled bed in these experiments were also sufficient to initiate vortex ripples, though, for the finer sands, these ripples always started at some point at the ends of the test bed. However, it appeared that if the test bed were boundless the criteria for grain motion and ripple formation would indeed be distinct. The difference between ¢$, and $, appears to be small, negligible for coarse sand, but increasing as the grain size (r) decreases. In these experiments, the rolling-grain ripples of Bagnold (1946), were observed with the 0.18- and 0.21-millimeter sands but not with the 0.55-millimeter sand. Such an unpredictable occurrence is characteristic of previous observations. Despite their persistence, these ripples appeared to play no part in the development of vortex ripples. This passive behavior contrasts with the more active role described by Bagnold (1946), Manohar (1955), and Carstens, Neilson, and Altinbilek (1969), and suggests that rolling-grain ripples are not an essential part of ripple development. 2. Growth of Ripples. The profiles were always initially two dimensional when the ripples were small, but during the subsequent period of rapid growth they became, in varying degree, three dimensional. Following this period, as the profiles approached equilibrium, the ripples either became again two dimensional or remained more or less three dimensional. The dimensionless initial ripple length, \;/D, was found to be approximately constant for each sand, to increase moderately as r decreased, and to be independent of a/D. That Xj is independent of a Le provides insight into the mechanics of ripple initiation, and suggests the involvement of stress and the distances grains move over the flat bed, rather than the action of a disturbance in the water, which would move with amplitude a. However, Carstens and Neilson (1967) mention vortices in the water just over the bed and perpendicular to the flow, apparently, before ripples appear. Also, Folk (1976), cites many examples to show that bed forms are the result, rather than the cause, < of vortices in the flow. Observations of growing ripples suggest that n/a and i/a depend primarily on (D/a)n, where n is the number of cycles since the ripples first appeared. These (D/a)n observations are restricted to the condition that U/U. % 1 and, except for three experiments, to the use of the 0.55-millimeter sand. The maturity of ripple profiles has been found to depend on (D/a)n, involving an interval of time propor- tional to (a/D)T. The period of rapid, and partly three-dimensional, ripple growth may have been lengthened by the presence of the channel walls. On an unbounded bed, the development of a partly three- dimensional bed form might be hastened by lateral spreading of more nearly stable forms from wherever they occur. 3. Size and Shape of Stable Ripples. Observations of i/D (Fig. 21) and of \/a and n/a (Figs. 22 and 23) in this study are in general agreement with most of the previous studies summarized in Figure 5. Despite minor differences, the observations have confirmed the results of Mogridge and Kamphuis (1972), and extended them to sands of different sizes at large a and T. The observations have thereby supported the validity of the approximate equation (9) which states that \ is proportional to a, and that, with a held constant, X is relatively insensitive to changes in T and to the associated changes in U. These remarkable results deserve further study and explanation. Occasional failures of laboratory observations to follow the curves of Mogridge and Kamphuis (1972) in Figure 5 might be explained as the effects of boundaries, by failure to achieve a steady state, or, possibly, as distortions associated with oscillating trays. However, field and some laboratory observations of A/D follow a trend below and away from these curves as a/D increases to large values. This trend was discussed in Section I,3,d and was identified with fine sands. It was then suggested that the plots of Mogridge and Kamphuis (1972) might require modification for smaller values of r. Whatever the effects of r, observed values of A/D should fall progressively below the curves of Mogridge and Kamphuis (1972) as a/D increases to large values. Continued adherence to equation (10) would produce comparably large values of X/D. As a _ becomes infinite, the oscillatory ripples cannot grow to infinite size but rather must turn 74 into ripples characteristic of steady flow. Such ripples are of limited length. Yalin (1977) cites the approximation, * = 1,000 (steady flow) (26) (and replaces the constant, 1,000, with a function of Reynolds number. Yalin's data have an average i/D of around 800). A relation of the form of equation (26), with the constant lowered, moderately, to between 500 and 600, is consistent with the trends of the observations of Inman (1957) and Dingler (1975) for large a/D (see Fig. 5,e and f). Then, for oscillatory ripples, if $ is held constant and both a and T become large, A/a must ultimately become small. Such behavior is not found in the plots of Mogridge and Kamphuis (1972) which, as T becomes infinite and Xo (eq. 7) approaches zero, require \/a to remain, approxi- mately, constant, as in equation (8). The transition from oscillatory to steady-flow ripples suggests an analogy with the transition from laminar to turbulent flow in a rough pipe. This analogy is illustrated by schematic plots in Figure 31. For pipe flow (Fig. 3la), as the Reynolds number, R, increases, a friction coefficient, C¢, breaks away from its laminar asymptote at the onset of turbulence ultimately to approach a constant value depending on the wall roughness defined by a parameter, k. For ripples (Fig. 31b), as a/D increases, \/D diverges from its oscillatory asymptote, ultimately to approach the constant value for steady flow. The transition has been presumed to depend on some function k(¢,r) which combines the effects of $ and r (possibly like U/w as discussed in Section I,3,d) and is analogous to the parameter k in pipe flow. In the case of pipe flow, transition to turbulent flow is enhanced and shifted to lower values of R_ by disturbances in the incoming flow such as slight turbulence or swirl. Extending the analogy, transition from oscillatory to steady-flow ripples might be enhanced and shifted to lower values of a/D by deviations from pure sinusoidal two-dimensional flow over the bed. The analogy then suggests that, for given a/D, 9, r, values of X/D would tend to be larger (closer to eq. 10) in a water tunnel, which provides a sinusoidal two-dimensional flow, and smaller (closer to eq. 26) on the sea bed where the flow is perturbed, tendencies which were observed in Figure 5 and discussed in Section I,3,d. Effects of perturbations in the flow might also account for some of the scatter found in plots of X/D and i/a, particularly from field observations. Results of this study (summarized in App. A) show that two- dimensional profiles, developing under the same given values of a and T and with minimum end constraints, attain nearly identical final forms, even when the initial bed forms were very different. This indicates that unconstrained equilibrium profiles are independent of 75 log C, Figure 31. (a) pipe flow turbulent (b) ripples steady, eq.(26) log = Analogy between (a) transition from laminar to turbulent flow in a rough pipe, and (b) transition from oscillatory-flow to steady-flow ripples. 76 their histories. However, in experiments 37 to 49 with fixed end crests (Fig. 25), there was a variety of ripple forms, corresponding to different values of a with a fixed value of A, within the limits of equations (24) and (25). In addition, when the profiles were strained beyond these limits, a hysteresis resulted (discussion of Figs. 17 and 18 in Sec. V,5). On the unbounded seabed, ripples might be expected to show strain while still adjusting to previous changes in a, though, presumably, the time of adjustment, and with it the maximum strain encountered, would be reduced by the freedom of the profile to extend and contract longitudinally. A quicker adjustment might result from the opportunity for lateral motions combined with devi- ations of the bed forms from two dimensionality. On the seabed examples where a crestline stops or branches into two are common. The lateral movement of such features into or out of an area could change the average ripple length almost continuously. This mechanism was alluded to in Section VII,2 in regard to the growth of ripples. 4. Two- and Three-Dimensional Bed Forms. Equilibrium bed forms were always two dimensional with the 0.55- millimeter sand but both two and three dimensional with the two finer sands. Attempts to further define the conditions for two- and three- dimensional forms are inconclusive. With the 0.18-millimeter sand, two-. or three-dimensional forms were observed, with fair consistency, as @$ -was less than or greater than 21.3 (U = 25.2 centimeters per second; Fig. 27). With the 0.21-millimeter sand the separation of two- and three-dimensional forms was not so simple. A suggestion for this difference in behavior between the two finer sands, based on grain-size distribution, was offered in Section VI,2. Generally, irregularities in the occurrence of two- and three-dimensional forms might stem from limitations of time and space. It was noted in Section V,1 that pro- files are prone to become three dimensional during periods of growth and that, with the finer sands, during slow or imperceptible growth, recurrence of two and three dimensionality was not uncommon. There remains then a possibility that some of the observed final bed forms might have modified or changed character if the experiments had been prolonged. Although the channel walls clearly suppress large three- dimensional motions, they also initiate small ones, since at a wall the vorticity vector of the flow must be parallel to the wall, rather than normal to it as in strictly two-dimensional flow. Once established, symmetrical slightly three-dimensional flows and associated bed forms might become unstable, lose symmetry and become increasingly three dimensional. Since such wall effects on the transition from two- to three-dimensional bed forms in these experiments are not known to be negligible, the curves in Figures 27 and 28 must be used with caution. After transition, the effects of the walls on the further development of three-dimensional bed forms must be presumed significant, and for this reason their final characteristics have not been described in detail. 77 Although a three-dimensional character of the bed form might be initiated and then conditioned by the presence of channel walls, such constraints are by no means its only possible source. This is shown by the three-dimensional ripples observed by Inman (1957) on the unbounded seabed. A likely source here are the lateral perturbations in the flow over the bottom which must be found under surface waves which are themselves not two dimensional. This supposition is consis- tent with the irregular occurrence of long-, medium-, and short-crested ripples (Fig. 30). It is further supported by the observed tendencies for long-crested ripples to occur in shallow water under shoaling waves and in protected areas where the fetch is short--both conditions likely to favor more regular and two-dimensional surface waves. However, the observations at short fetch are also at smaller values of a/D. It has already been noted that oscillatory ripples should become more like steady-flow ripples as a and T become large. Raudkivi (1976; pp. 62-65) describes steady-flow ripples as typically three dimensional. These considerations are consistent with a tendency for oscillatory ripples to become more three dimensional with increasing a/D. ce Disappearance of Ripples. To forestall damage to the screens (Table A-2, experiment 67) in these experiments, 9 had to be limited to values unable to produce sheet flow. Therefore, this section only discusses the criteria of Manohar (1955), Chan, Baird, and Round (1972), and Dingler (1975), given by equations (12), (13), and (14), respectively. In equations (12) and (13), 95 depends on both a/D and r; in equation (14) this dependence is absent, perhaps due to relatively small ranges of a/D and r in Dingler's (1975) observations of sheet flow, with a/D between 5x10° and 10+ and with D between 0.128 and 0.158 millimeters, corresponding to r ~5.8 and 8.0. Curves of $ = d<, given by equations (12) and (13) for r = 5, 10 and by equation (14), are plotted against a/D in Figure 32 (discussed in Sec. VII,6). The curves in Figure 32 are solid where supported by observations and dashed where extrapolated. In the region of Dingler's (1975) observations of sheet flow, the criteria are in remarkable agreement, considering their diverse derivations and the distortions to which equations (12) and (13) are ostensibly subject. Unnatural accelerations of the oscillating tray may have reduced the ¢, observed by Manohar (1955), especially at smaller values of a/D. Effects of confinement in the pipe test section of Chan, Baird, and Round (1972) are clearly severe on bed forms with height comparable to the pipe diameter (5.1 centimeters), but may have lessened as the bed flattened to approach sheet flow. (However, in experiment 55 $ exceeded $, by eq. 13 without attaining sheet flow.) Taken together, equations (12), (13), and (14) suggest a $, as a modifica- tion of equation (14) to allow for relatively slight effects of a/D and r. These effects are not yet established, but $, appears to increase with increasing a/D and, usually, with decreasing r, though perhaps not always, as suggested by the crossing extrapolations of equation (12) FOr, =e landeliOr 78 2 5x!l0 VT Vaasa! aan ae al | Tasmnelimua "malian ig Tec vA 4 f ae sheet flow a oe Ee, eae moe fT a ripples sheet flow Inman (1957) Dingler (1975) ola Figure 32. Limits to the occurrence oi ripples on the seabed (data from inman, 1957, and Dingler, 1975). us 6. Occurrence of Ripples. With the criteria of Manohar (1955) and Dingler (1975) providing a maximum above which ripples disappear, and with the "composite" curve for ¢, in Figure 2, together with the remarks in Section VII,1 re- garding a minimum ¢ below which ripples cannot be sustained in equilib- rium, an attempt may be made to map out the domain where ripples occur. Observations of ripples and of sheet flow in the field by Inman (1957) and Dingler (1975) are plotted in Figure 32 according to their values of a/D and 9. The criteria for sheet flow have been drawn and the composite curve for oo as function of a/D has been transcribed from Figure 2. A lower bound for equilibrium ripples, which is poorly defined, is here suggested as » = ¢c/5, which lies along the lower edge of the field observations just below the $./4 suggested by Lofquist's (1975) laboratory study, and somewhat farther below the point from Carstens, Neilson, and Altinbilek's (1969) experiment 62 which is marked by an arrow. A few observations that fall below the curve are to be expected as representing cases of relict ripples formed by previous and stronger flows. It is interesting that more than half of the field observations of ripples are for flow conditions too weak to move sand on a flat bed according to the composite curve for laboratory data in Figure 2. Also, for about half of the experiments in this study values of 9 were below de> and the ob- served sand motion showed that the profiles were not relict but were in equilibrium with the flow. The upper-bound criterion of Dingler (1975) is, of course, in accord with his own and Inman's (1957) observations of sheet flow; the criteria of Manohar (1955) and of Chan, Baird and Round (1972), represented by two selected values of r, are also in fair agreement. Although 4 and a/D are independent variables, the distribution of the points of ripple observations in Figure 32 shows a trend of @ with a/D with hardly more scatter than is often tclerated in interpret-. ing functional relationships. In this case, 9 increases roughly as the square of a/D. This trend reflects a limited range of T in the field observations of Inman (1957) and Dingler (1975). For two-thirds of these observations, T fell in the range from 8 to 12 seconds. This range is characteristic of the California coast and may not appear elsewhere. The observed trend of 9 with a/D is then not a functional relationship but a circumstantial correlation. It makes separation of the effects of and a/D upon A/D or n/D very difficult. - With po /P regarded as constant for natural sands, the occurrence of ripples, as well as other of their characteristics, depends on the three independent variables a/D, ¢, and r. The choice of a/D, rather than r, as abscissa in Figure 32 owes much to the fact that the trend of ¢,. with a/D is fairly clear while the effects of r on $¢ remain contradictory and obscure. (In field data, r depends primarily on D.) Similarly, \4/D and n/D have been found to depend more 80 strongly and clearly on a/D than on either $» or r. Figure 32 may be compared with Figure 9 in Inman (1957) which attempts to show the domain of ripple occurrence on a field of U versus D. With the present dimensionless variables, this is roughly equivalent to $2 versus 72/3, In Inman's plot the effects of a/D (or of T) are not indicated; in Figure 32 r is absent, except as a parameter in the criteria for sheet flow of Manohar (1955) and Chan, Baird, and Round (1972). VIII. SUMMARY This study has discussed observations of ripples from a series of laboratory experiments, without any attempt at theoretical models; the direct results of these observations are summarized in this section. A guide to this summary is provided by the list of “emerging questions" in Section I,3,h. These questions and related findings are put in two groups. The first group of findings relates to the initiation of grain motion and ripples on a flat bed. Their primary application is in support of physical models of ripple development: (a) ¢ has been found to be a function of both a/D and r (Fig. 12). (b) A prior compression of the sand surface was found to have only a small effect upon $9, (Fig. 12). (c) A Shields criterion for the initiation of grain motion under oscillatory flow was found in need of further modification (Fig. 13). (d) oy; was found to be distinct from $,, but the difference was small and detectable only with the finer sands (the arrows in Figs. 10 and 11). (e) Rolling-grain ripples were observed only with the two finer sands. These ripples appeared not to play an active or necessary role in the formation of vortex ripples. (f) The initial length of vortex ripples was found to depend slightly on D but not on a (Fig. 15). This independence suggests that vortex ripples are not caused by a prior disturbance moving with the water. The second group of findings relates to the character of bed forms, whether in growth or in equilibrium. In large part, they con- firm the design curves of Mogridge and Kamphuis (1972), while enlarging their experimental base with other sand grain sizes at moderate and large values of a and T. 8 | Beyond this: (g) The growth of n/a and X/a, with time, has been found to be a function of (D/a)n. This function shows ripples to be mature when n > a/D (Figs. 19 and 20). (h) The variety of initial bed forms which converge into final forms depending, essentially, only on a and T has supported the conclusion that an equilibrium profile, when unconfined, is independent of its history (Figs. 21, 22, and 23). (i) When confined at its ends, an equilibrium profile has been found to have various sizes and shapes determined by its value of A/a, which is contained within definite limits but is otherwise determined by the history of the profile (Fig. 25). (j) Both two- and three-dimensional final bed forms were observed. Clear criteria for these forms were not established, but a dependence on @ and, possibly, on a/D is apparent (Figs. 27, 28, and 29). Three dimensionality was found enhanced by a reduction in D and by growth or instability of the profile. In this study, more than usual, discussions have involved the find- ings of earlier studies, and it is hoped that the attempts to relate and interpret them (e.g., Figs. 3, 4, 5, 30, and 32) have a value apart from the new laboratory observations. More work is needed to improve pre- diction of the rippled character of the seabed under given conditions of flow. The design curves of Mogridge and Kamphuis (1972), useful as they are, need modification at large values of a/D as shown by the field observations of Inman (1957) and Dingler (1975) and as discussed in Section VII,3. Conditions for two- and three-dimensional bed forms re- main poorly defined (Secs. VI,2,3 and VII,4). Despite these reservations, the character of the seabed can even now be predicted well enough to encourage studies of the bottom stress and of the amount of sand in sus- pension, both of which depend on the bed forms. As described in Section I,1, results of these studies find application in coastal engineering problems of surface wave decay and of sand transport. 82 LITERATURE CITED BAGNOLD, R.A., "Motion of Waves in Shallow Water, Interaction Between Waves and Sand Bottoms," Proceedings, Royal Soctety of London, Series A, Vol. 187, 1946, pp. 1-18. CARSTENS, M.R., ''Similarity Laws for Localized Scour,' Journal of the Hydraulics Divistcn, American Society of Civil Engineers, Vol. 92, No. HY3, May 1966, pp. 13-36. CARSTENS, M.R., and NEILSON, F.M., “Evolution of a Duned Bed under Oscillatory Flow," Journal of Geophysical Research, Vol. 72, No. 12, June, 1967, pp. 3053-3059. CARSTENS, M.R., NEILSCN, F.M., and ALTINBILEK, H.D., "Bed Forms Generated in the Laboratory Under an Oscillatery Flow: Analytical and Experimental Study,"’ TM-28, U.&. Army, Corps of Engineers, Coastal Engineering Research Center, Washington, D.C., June 1969. CHAN, K.W., BAIRD, M.H.I., and ROUNT, G.F., "Behaviour of Beas of Dense Particles in a Horizontally Oscillating Liquid," Proceedings, Royal Soevety of Lonaon, Series A, Vol. 330, 1972, pp. 537-559. COLLINS, J.1., and CHESNUTT, C.B., "Grain Shape and Size Distribution Effects in Coastal Models," TP 76-11, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1976. DINGLER, J.R., 'Wave-Formed Ripples in Nearshore Sands," Ph.D. Dissertation, Department of Oceanography, University of California, San Diego, Calif., 1975. FOLK, R.L., "Rollers and Ripples in Sand, Streams and Sky: Rhythmic Alteration of Transverse and Longitudinal Vortices in Three Orders," Sedimentology, No. 23, Oct., 1976, pp. 649-669. HOMMA, M., and HORTKAWA, K., ''Suspended Sediment due to Wave Acticn," Proceedings of the Etghth Coastal Engineering Conference, 1962, pp. 168-193. HORIKAWA, K., and WATANABE, A., "A Study on Sand Movement due te Wave Action,"' Coastal Engineering tr. Japan, Vol. 10, 1967, pp. 39-57. INMAN, D.L., "Wave-Gernerated Ripples in Nearshore Sands,"' TM-100, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington D.C., Oct. 1957. INMAN, D.L., and BOWEN, A.J., ''Flume Experiments on Sand Transport by Waves and Currents," Proceedings of the Eighth Coastal Engineering Conference, 1962, pp. 137-150. 83 JONSSON, I.G., "Wave Boundary Layers and Friction Factors," Proceedings of the lOth Coastal Engtneertng Conference, American Society of Civil Engineers, 1966, pp. 127-148. KAMPHUIS, J.W., "Friction Factor under Oscillatory Waves," Journal of the Waterways, Harbors and Coastal Engineering Divtston, American Society of Civil Engineers, Vol. 101, No. WW2, May 1975, pp. 135-144 KENNEDY, J.F., and FALCON, M., ''Wave Generated Sediment Ripples," ‘Report No. 86, Department of Civil Engineering, Hydrodynamics Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., Aug. 1965. KOMAR, P.D., and MILLER, M.C., "The Threshold of Sediment Movement Under Oscillatory Water Waves," Journal of Sedimentary Petrology, Vol. 43, No. 4, Dec. 1973, pp. 1101-1110. KOMAR, P.D., and MILLER, M.C., "The Initiation of Oscillatory Ripple Marks and the Development of Plane-Bed at High Shear Stresses under Waves,'' Journal of Sedimentary Petrology, Vol. 45, No. 3, Sept. 1975, pp. 697-703. LOFQUIST, K.E.B., “An Effect of Permeability on Sand Transport by Waves," TM-62, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Dec. 1975. LOFQUIST, K.E.B., "A Positive Displacement Oscillatory Water Tunnel," MR 77-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Feb. 1977. MADSEN, 0.S., and GRANT, W.D., ''The Threshold of Sediment Movement Under Oscillatory Waves: A Discussion," Journal of Sedimentary Petrology, Vol. 45, No. 1, Mar. 1975, pp. 359-367. MADSEN, O.S., and GRANT, W.D., "Sediment Transport in the Coastal Environment ,"' Report No. 209, Department of Civil Engineering, Hydro- dynamics Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., Jan. 1976. MANOHAR, V., ''Mechanics of Bottom Sediment Movement Due to Wave Action," T-75, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., June 1955. MOGRIDGE, G.R., and KAMPHUIS, J.W., "Experiments on Bed Forms Generated by Wave Action," Proceedings of the 13th Coastal Engineering Conference, Vol. II, 1972, pp. 1123-1142. NIELSEN, P., '' A Note on Wave Ripple Geometry,'t Progress Report No. 43, Institute of Hydrodynamics and Hydraulic Engineering (ISVA), Technical University of Denmark, Aug. 1977, pp. 17-22. 84 RANCE, P.J., and WARREN, N.F., "The Threshold of Movement of Coarse Material in Oscillatory Flow," Proceedings of the llth Coastal Engtneertng Conference, American Society of Civil Engineers, 1968, pp. 487-491. RAUDKIVI, A.J., Loose Boundary Hydraulics, 24 ed., Pergamon Press, Elmsford, N.Y., 1976. SCOTT, T.S., "Sand Movement by Waves,'' TM-48, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington D.C., Aug. 1954. SHIELDS, A., "Application of Similarity Principles and Turbulence Research to Bed-Load Movement," translated from "Anwendung der Aenlichkeitsmechanik und der Turbulenz forschung auf die Geschiebebewegung,'' Mitt. d. Preuss. Versuchsanstalt f. Wasserbau u. Schiffbau, Berlin, Heft 26, 1936, by W.P. Ott and J.C. van Uchelen, U.S. Dept. of Agriculture, Soil Conservation Service Cooperative Laboratory, California Institute of Technology, Pasadena, Calif., 1936. SHULYAK, B.A., "Certain Aspects of the Interaction Between Wave Flow and a Deformable Bottom at Low Velocities," Dynamics and Morphology of Sea Coasts, translation of Dinamtka t Morpnologtya Morskikh Beregov, Moskva, 1961, available at National Technical Information Service, Springfield, Va., 1969, pp. 218-283. SILVESTER, R., Coastal Engineering, Vol. 2, Elsevier, Amsterdam, 1974. STERNBERG, R.W., and LARSEN, L.H., "Threshold of. Sediment Movement by Open Ocean Waves: Observations,'' Deep-Sea Research, Vol. 22, 1975, pp. 299-309. U.S. ARMY CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, 34 ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. | YALIN, M.S., "On the Determination of Ripple Length," Journal of the i Hydraulics Division, American Society of Civil Engineers, Vol. 103, | No. HY4, Apr. 1977, pp. 439-442. YALIN, M.S. and RUSSELL, R.C.H., "Similarity in Sediment Transport due to Waves," Proceedings of the Eighth Coastal Engtneertng Conference, 1962, pp. 151-167. 85 7h eee a fr { ino he a CAMPS te ts Cire at m EMMETT Nis facia Roa ree \ } Mi eae Syste ie Dit ‘aapwilte abait ‘Mas PeLezoKDs LSP rot: vane Me é wa Hel Pik m ak hihi i and Ret. eh ay af curesny iaigeerrit "AG¥ od, Lobns rh we AveD Asma cin aes oo) Leet ae damyrniinneOiahka® nce reo uh aia ‘Sanne me ae: Wrriees ‘3 Ri artes ees aye Ly OR: rire psig ety! Ws ee Te) Fe earner fan at a ite ie’ ‘om | soul efi eenod sas hs fas nih Bits Sa are, meee aes drtean Sein oo wy ae ne 4 AK rsh ie mt) ‘tant of “Teh tgin ° Nias OOK S. DRS Ce op ‘af tncknology. s Sano tor Tpite al or ggi8 26 a0 , pu niahoae ent oy" T SOL ofoW) ok epoendnne livid to (relied wee brems eohedaed Eee Ben nent by alba FR Twa! ’ i sb hirp ion “i ana , ee 4 epbaut beta” me ; é; it a ‘ubtaand ae Sad: tah ako ony PAA oe ae a ON eh le x fet 0 4, Bae te! ue Ty r AMEN ft is oh. 7 fab inert inomet Wer he cle nee by. We on, Proadsdirae oromia: Dek"! haa ings eh wig afore, WOE. LD Lae. we Ee iiad 1 ae i 7 nt Wea rriprort. Peper Noy Poet! th niyeeaitc w., ing tote tig: ae ij 7 hy De a 8 ALE ee ba iy, APPENDIX A DATA FROM 104 EXPERIMENTS ON RIPPLE GROWTH AND EQUILIBRIUM FORMS Tables A-1, A-2, and A-3 contain data and notes for experiments with the 0.55-, 0.18-, and 0.21-millimeter sands. The experiments are listed chronologically. Special measures carried out between experiments are noted in proper sequence. The value of N shows the stroke for each experiment. Experiments with the bed initially leveled are identified by footnotes and by observations of \;, which pertain only to these experiments. Absence of values for X,n indicates a final bed form too three dimensional for iX,n to be defined. Included with some experiments are notes on their progress or outcome. These refer mostly to the two- or three-dimensional character of final bed forms with the 0.18- and 0.21-millimeter sands. (With the 0.55-millimeter sand, final bed forms are all two dimensional.) When more than one of the three T, T,, Ty. are entered, the first listed is that at which the ripple profile was allowed to develop. a = 0.4576 N centimeters j=} uN 2.87 N/T centimeters per second 87 Table A-1. Data and notes for experiments with the 0.55-millimeter sand (r = 52.39). Expt. No. a T,T, G nN n U,U, $b Duration (cm) (s) (cm) (cm) (cm) (cm/s) (hr:min) ir ee at liteestoa | 8. | 8.08(c) | aes 33 35. | 35.6(¢) | 13. | 13.95¢c) | 18.8 . 5 moo 12 (c)) 13.68(c) 1:00 iad Ea 36.6 5.50 42.7 8.90 43.4 20.76 3:37 35.3 43.1 20.50 1:56 0 32.04 3 13.66 8.42 14 46.8 33.3(c) 12.25(c) 3:47 9 23.370 er al ae ee eee 15.01 ee N co co Sand in one spool redistributed manually Brass crests in spools shifted to 22 cm from screens rae ES lexperiments with bed initially leveled. 88 Table A-1. Data and notes for experiments with the 0.55-millimeter sand (r = 52.39) .-Continued. Expt. No. a Duration (cm) ie ao ey i) (hr:min) 11.16 13.12 Brass crests mounted at ends of sand bed atop the curved ramps a w a Nn _ w la N i=} > N = w N = 0.35 Sa a 18.18 Le (Sea Ee a ee ae es Brass crests removed 51! 18.3 3.42 DoS 27.3 4.13 33.6 12.46 3.58(c) 32.1(c) 11.37(c) Surface leveled and compressed 1.5 mm C= | « [ws | wole | se] os | ono] woo] a | Sand in one spool reduced and redistributed 6 0 39.7 17.33 Sand redistribution has small effect Surface leveled and compressed 0.83 mm 2.78 31.0 a 61 30.7(c) 10.39(c) 2.81(c) Experiment interrupted and sand added to spools lExperiments with bed initially leveled. 89 Table A-2. Data and notes for experiments with the 0.18-millimeter sand (r = 9.81). Expt. No. T,Ty,Ty on U,Uc Uy bsdcror Duration ) (s) eS (cm/s)” (hr:min) int: aad ela = =F PP l= packs = ee Final bed form: 3D Final bed form: trend to 2D w > w i=) w 30 trend to 3D form: 16.03 0 10.84 form: trend to 2D 12.73 form: mostly 2D 10.99 form: mostly 3D w w N a an “ _ “ , ~ oo “N N o T reduced in steps from 7.26 s to 4.93 s where screen tears from seal. Final bed form is mostly 3D but with two large rounded crests formed by sand exchange with screens; not entered in Figure 27. 2.97(c) Final bed form: 2D, 3D Final bed form: 2D Final bed form: mostly 3D Final bed form: mostly 3D N w I Nn o oo lExperiments with the bed initially leveled. 2Final bed form too three dimensional to determine A,n. 90 Table A-2. Data and notes for experiments with the 0.18-millimeter sand (r = 9.81).-Continued. T,TosTy Aj r n U,Uc Uy osbcoby Duration (s) (cm) (cm) (cm) (cm/s) (hr:min) 721 : 20 2.71(c) 11.5 1.69 15.16(c) Final bed ; 3.03 11.5 2 : 21.6 28.3(c) 33.6(r) form: 2D, following long development with sporadic 7.58 0 . 22.8 5.44(c) 31.7(c) 33.86(c) 4.53(r) 38.1(r) 48.82(r) form: 2D, following long development with sporadic bed form: mostly 3D 2.56(r) 0 28.1(r) 26.55(r) 2.96(c) 24.3(c) 19.91(c) Final bed lexperiments with the bed initially leveled. 2Rinal bed form two three dimensional to determine Ayn. 9I Table A-3. Data and notes for experiments with the 0.21-millimeter sand (r = 13.25). Expt. No. a T,To,Ty G d n U,U.,U, Duration (cm) (s) (cm) (cm) (cm) (cm/s) (hr:min) g5l 40 18.3 4.01(c) 28.7(c) 23.74(c) Final bed form: 2D, 3D 22.7 23.0 15.27 Final bed form: 2D 9 3.22 25.6 18.85 Final bed form: 2D 88 40 18.3 28.8 Final bed form: 2D, 3D 89 40 18.3 3.57 2.67 - 32.2 Final bed form: mostly 3D 90! 25 11.4 2.91(c) 2.18 24.7(c) 17.61(c) Final bed form: 2D 91 25 11.4 2.57 ileal Final bed form: mostly 2D 92 25 11.4 2.23 13.3 29.98 Final bed form: 2D 93 25 11.4 Final bed form; mostly 2D 94 25 11.4 1.80 11.8 1.74 39.9 Final bed form: 2D, crest ridges yaw alternately ; 95 25 11.4 Final bed form: 2D 961 30 kos 3.33(c) 5.36 25.9(c) 19.36(c) Final bed form: 2D 97 30 13.7 2.70 5.3 Final bed form: mostly 3D 98 30 DSi) 2.40 14.5 37.28 Final bed form: mostly 2D, following a 3D regime 99 30 13.7 Final bed form: mostly 3D Sand accumulated in spools is spread over the ends of the bed 100 30 13.7 2.71 Final bed form: mostly 2D 4.59(r) 5.70 37.6(r) 40.76(r) 5.50(c) 31.4(c) 28.40(c) Sand accumulated in spools is spread over the ends of the bed Final bed form: mostly 3D 102 60 27.5 Final bed form: mostly 2D 6.02(r) 6.71(c) 38.2(r) 34.3(c) 42.14(r) 33.91(c) Final bed form: mostly 3D Sand accumulated in spools is spread over the ends of the bed 104 80 36.6 9.66 7:23 Final bed form: mostly 2D lExperiments with the bed initially leveled. 2Final bed form too three dimensional to determine \,n. 92 APPENDIX B PHOTOS OF INITIAL AND FINAL BED FORMS FOR THE 24 EXPERIMENTS STARTING FROM AN INITIALLY LEVELED BED 93 Figure B-1. Initial and final bed forms for experiments with the 0.55-millimeter sand. Experiment numbers shown at left of each pair of profiles. 94 Oe) Figure B-1. Initial and final bed forms for experiments with the 0.55-millimeter sand. Experiment numbers shown at left of each pair of profiles.--Continued 95 Figure B-1. Initial and final bed forms for experiments with the 0.55-millimeter sand. Experiment numbers shown at left of each pair of profiles.--Continued 96 58 59 62 Figure B-2. Initial and final bed forms for experiments with the 0.18-millimeter sand. Experiment numbers shown at left of each pair of profiles. 97 Figure B-2. Initial and final bed forms for experiments with the 0.18-millimeter sand. Experiment numbers shown at left of each pair of profiles.--Continued 98 8! 82 84 Figure B-2. Initial and final bed forms for experiments with the 0.18-millimeter sand. Experiment numbers shown at left of each pair of profiles.--Continued 99 85 90 96 Figure B-3. Initial and final bed forms for experiments with the 0.21-millimeter sand. Experiment numbers shown at left of each pair of profiles. 100 Ke} 103 Figure B-3. Initial and final bed forms for experiments with the 0.21-millimeter sand. 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