ws, Av ahr ae QaaTe € re TP 77-11 Forces Exerted by Waves on a Pipeline At or Near the Ocean Bottom by George L. Bowie TECHNICAL PAPER NO. 77-11 OCTOBER 1977 Approved for public release; distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER Kingman Building 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. /WWHO1 MBL A YW Bqd7d 5 il 5 0301 00 | UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS T. REPORT NUMBER 2. GOVT ACCESSION NO,| 3. RECIPIENT'S CATALOG NUMBER TP 77-11 3. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Technical Paper 6. PERFORMING ORG. REPORT NUMBER Technical Report HEL 9-24 8. CONTRACT OR GRANT NUMBER(s) FORCES EXERTED BY WAVES ON A PIPELINE AT OR NEAR THE OCEAN BOTTOM 7. AUTHOR(s) DACW72-74-C-0004 George L. Bowie 10. PROGRAM ELEMENT, PROJECT, TASK PERFORMING ORGANIZATION NAME AND ADDRESS AREA & WORK UNIT NUMBERS 9. University of California Hydraulic Engineering Laboratory Berkeley, California 94720 CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CEREN-DE) Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) F31234 REPORT DATE “october 1977 NUMBER OF PAGES ‘177 15. SECURITY CLASS. (of this report) UNCLASS IF TED 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release; distribution unlimited. DISTRIBUTION STATEMENT (of this Report) 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) Submarine pipeline Two- and three-dimensional experiments Wave force analysis ig Wave-induced lift forces ABSTRACT (Continue on reverse side if necesaary and identify by block number) The wave-induced forces on a submarine pipeline near the ocean floor consist of several components--inertial forces, drag forces, lift forces, and under some conditions, eddy-induced forces. For a pipeline touching the bottom, or at a small clearance above the bottom, the lift force is the predominant force in the vertical direction. This force is generally expressed as Fy, = 1/2 Cy p A u*, and is added as a lift term to the Morison equation. (Continued) DD , eee 1473. EDITION OF ft Nov 6515S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Dats Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) The experimental results of this investigation, however, show that this steady-flow lift model is inadequate for wave-induced oscillatory flows. For pipelines at small clearances above the bottom, viscous effects near the bottom clearance constriction may result in lift forces acting in both the upward and downward directions during different parts of the wave cycle. In addition, the maximum positive and negative lift forces may not correspond to the positions of maximum horizontal velocities in the wave cycle. A modified lift force model of the form, F;, = 1/2 Cy p A tinesc” [cos? (6-6) - k], is proposed where the parameters, Cy, 9, and k, may vary accordingly to allow adequate description of all characteristics of the lift force phenomenon. Quantitative relationships between these unknown lift force parameters and various dimensionless parameters defining the wave and pipe conditions were found. These relationships exhibited good correlation for all wave conditions, bottom clearances, pipe diameters, and orientation angles. 2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report is published to provide coastal engineers with an analysis of wave-induced forces on a submarine pipeline near the ocean floor. The work was carried out under the structural design program of the U.S. Army Coastal Engineering Research Center (CERC). The report was prepared by George L. Bowie, Research Assistant, University of California, Berkeley, under CERC Contract No. DACW72-74- C-0004. The author acknowledges the help and advice of Professor R.L. Wiegel, Professor J.W. Johnson, and Dr. J.D. Cumming, throughout different stages of the project. Many thanks also go to Dr. M.F. Al-Kazily, J. Allison, and L. Magel for their help in designing and setting up the instrumentation for the experiments, and to W. Krogmoe, E. Parscale, and W. Matthew for their skillful assistance in building the models. Dr. J.R. Weggel was the CERC contract monitor, under the general supervision of G.M. Watts, Chief, Engineering Development Division. Comments on this publication are invited. Approved for publication 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 CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) SYMBOLS AND DEFINITIONS . I WAVE FORCE ANALYSIS . I. 2. Wave-Induced Lift Forces 3. Model for Wave-Induced Lift Fomees 4. Extension of Model to Higher Order Theories, II EXPERIMENTAL INVESTIGATION 1. Experimental Equipment . Bo 2. Procedure for Two-Dimensional - Experimentse 3. Procedure for Three-Dimensional Experiments 4. Data Reduction ..- WICH RESULTS AND DISCUSSION : : 1. Resultant Force Through Wave Gyele : 2. Orientation Angle Considerations i 3. Interrelationships Between C,;, >, and ie 4. Relationships Between $ and k and Parameters Defining the Wave and Pipeline Conditions 0 é 5. Relationships Between » (clear/Dia) and k (clean) Dia) and Parameters ame the Wave and Pipeline Conditions. : é 6. Relationships Beeweon the Coefficients oe Lift ad Parameters Defining the Wave and Pipeline Conditions. 7. Relationships Between the Lift Forces and Parameters Defining the Wave and Pipeline Conditions . : 8. Relationships Involving the Vertical Coefficients of Mass and Drag and the Vertical Inertial and Drag Forces 9. Relationships Between “the. Horizontal Coefficient oF Mass and Parameters eee the Wave and Pipeline Conditions : 10. Relationships Involving the ‘Homizoncal Cost clen of Drag . Sistas 11. Example Preabilems IV CONCLUSIONS . V RECOMMENDATIONS FOR FURTHER RESEARCH Wave Force Components on 7 Pipelines Near the Bottom . LITERATURE CITED. Page CONTENTS APPENDIX Page A EEASI SQUARES MANATYVSI Se OR EXPERIMENTAL DATVAN eit eiivell eile) sian iui. B COMPUTER PROGRAM FOR VERTICAL LEAST SQUARES ANALYSIS (CTHLOSDIMENSUONVAG MUA) SS 66's 68 bo el oe OE lo Sydeo le Geld 140) C COMPUTER PROGRAM FOR VERTICAL LEAST SQUARES ANALYSIS (CUFINEISSDIONISINS TONAL, DMN) So 6 5 6 615 6 6 6 6.5 0 od 9 oo a, ad D COMPUTER PROGRAM FOR HORIZONTAL LEAST POOREST ANALYSIS (HWO=DEMENS TONAL DADA) 935) ee a Sea LAS E TABULATED VERTICAL FORCE DATA FROM TWO-DIMENSIONAL EXPERIMEN MS Isa suite meet FcR TAR et SAP OR ieee Mas ee Shs eee pa MOS F TABULATED VERTICAL FORCE DATA FROM THREE-DIMENSIONAL EXBE RUMEN TSH eee Mecca tein (ne Be Aaa Wenn OAM airs Resins a aces tech LO® G TABULATED HORIZONTAL FORCE DATA FROM TWO-DIMENSIONAL EER UINIEN SR meatal aun nie bm rorar iss vo Rine Mccuenul fore sive takcsh Jon Mura aa ecu eras acc et acontae LGA) TABLE Esteimatedmaccunacya Of experimenitealasmeasunce meme Siva eter eee ea ieuO FIGURES l. Chamee. iin IAs Waleln dimenSasiimg WeloOciey oc 4 6/6 6 6,6 6 5.5.0 5 2 0 All 2 CheMmeS im WssFe ioiceS Walin weSSiily wee CeeSte oo 5 co o 50 5 6 cic 0 5 Jhiske stores jonSMOMeEMOM 5» e 6% 6 oe 6 4 o 6 0.500 co o 6 oo o 6 596 2S AW Change in’ lett force recond for inexeasinig, bottom eliearantcer =) -) 7-5) 627 5 DSsctinneom SkSEC Nc aioe 2 6b ee cd So ooo lo be oo Gre 6 6 8 © Datliatlo@m @ic Maske sc@weS jORANSTOCS 5 co ¢ aio o%6 6 oo 6 6 oo o 6 Sil 7 Comparison of linear and Stokes' third-order theories: Simultaneous shift of lift force record as ¢ increases from 0° to 90° and k increases from 0 to 1 with increasing bottom clearance .....- 34 8 Comparison of lift force extreme cases for linear and Stokes! third- Gaels TEOMA Lee Parkes aun nya momen Ma) oa iapmele ohiotn chase Couette te eet se MEar et eS Q JFOIES TSEC HING SUIoysowies oe. BY si enh, TN eleiye Bo We oe She Se VE Sts 25 26 Bu 28 29 CONTENTS FIGURES --Continued Four-inch cylinder mounted in the flume. . . brie Test section (force meter) Test section mounted in position Test section and force transducer . Schematic of pipeline model Pipeline model Brush recording instruments Digitizer and recording instruments Calibration method for two-dimensional experiments Experimental arrangement for two-dimensional tests Calibration method for three-dimensional experiments Experimental arrangement for three-dimensional tests Definition sketch for three-dimensional experiments Example of data record Example of computer output for vertical least squares analysis Example of computer output for horizontal least squares analysis Resultant force through wave cycle for 0.001-foot clearance, 1.85-second period, and 0.24-foot height Resultant force through wave cycle for 1/16-inch ciearance, 1.86-second period, and 0.24 foot height . Resultant force through wave cycle for 1/8-inch clearance, 1.85-second period, and 0.25-foot height Resultant force through wave cycle for 3/16-inch clearance, 1.85-second period, and 0.25-foot height 60 61 62 63 64 65 30 31 32 33 34 35 36 Si 38 39 40 41 42 43 44 45 46 47 48 49 CONTENTS FIGURES-- Resultant force through wave cycle 1.86-second period, and 0.25-foot Resultant force through wave cycle 1.86-second period, and 0.24-foot Resultant force through wave cycle 1.86-second period, and 0.25-foot Change in resultant force with inc Resultant force through wave cycle Continued for 1/4-inch clearance, height . for l-inch clearance, height for 2-inch clearance, nenvehieae reasing clearance for l-inch clearance, 1.23-second period, and 0.3-foot height Resultant force through wave cycle 0.95-second period, and 0.24-foot Resultant force through wave cycle 0.95-second period, and 0.24-foot Resultant force through wave cycle 0.96-second period, and 0.25-foot Alternative approaches for handling pipeline orientation angles @ versus k Cj, versus k . Ci, versus @ . MATSCEIVE OSES COCimel Eile Ore Effective positive coefficient of Effective negative coefficient of Effective negative coefficient of Cy G1-k) or Cj, (k)) versus k - @ versus (clear/u,.,T) for 4-inch max @ WErsus (CleEie/Upesl)) were Sinica ® Wersus (Clese/Mesd)) wor Asie for 0.001-foot cle irance, height . for 1/16-inch clearance, height for 2-inch clearance, height lift versus k . lift versus @ . lift versus k . lift versus o . diameter <:. =. .« diameter diameter Page 66 67 68 69 Al 72 YS 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 k 0 versus versus versus versus versus versus versus versus versus CONTENTS FIGURES--Continued (clear/u,.,1) for 4-inch diameter ... . max (clear/upaxT) for 3-inch diameter (clear/umaxT) for 2-inch diameter (UnaxT/Dia) (Umax! /Dia) (Unaxclear/v) for 4-inch diameter (Umaxclear/v) for 4-inch diameter (clean / urn) atau) (clear/ujg,T) (Dia/u,,,T) k(clear/Dia) versus (clear/upgxT) (clear/Dia) p(clear/Dia) versus (clear/UmaxT) (clear/Dia) k(clear/Dia) versus vDia/u,4,!F (clear/u,,,T) (clear/Dia) o(clear/Dia) versus VDia/u,,4,T (clear/upg,T) (clear/Dia) Maximum lift force (positive or negative) versus the Reynolds Comparison of the horizontal Cy with potential flow theory for a flow with constant acceleration number . Horizontal Cy versus (clear/u,,.1) Page 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: SSS = Multiply by To obtain inches 25.4 millimeters ie 2.54 centimeters square inches 6.452 square centimeters cubic inches NOE SE) cubic centimeters feet 30.48 ceiutimeters 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 I SSSZ kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars 10197 = Wo" * 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) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! Ivo obtain Celsius (C) temperature readings from Fahrenheit (F) readings, WSS atommulas GC = (5/9) (2 382))5 To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. SYMBOLS AND DEFINITIONS projected area of pipe section orientation angle with respect to wave crests coefficient of drag coefficient of lift coefficient of transverse force due to eddy shedding bottom clearance coefficient of mass stillwater depth pipe diameter total wave-induced force drag force horizontal component of drag force vertical component of drag force horizontal component of total wave force calculated horizontal force at position o. in wave cycle inertial force horizontal component of inertial force vertical component of inertial force IIE iOreS transverse “lift'' force due to eddy shedding observed horizontal force at position oy in wave cycle observed vertical force at position Oo. in wave cycle vertical component of total wave force calculated vertical force at position oy in wave cycle u max SYMBOLS AND DEFINITIONS--Continued wave height negative fraction of lift force cycle wavelength wave period time since last wave crest passed over center of pipe section horizontal component of water particle velocity if pipeline was absent maximum horizontal water particle velocity if pipeline was absent volume of fluid displaced by pipe section vertical component of water particle velocity if pipeline was absent maximum vertical water particle velocity if pipeline was absent vertical distance of center of pipe section above bottom horizontal component of water particle acceleration if pipe- line was absent vertical component of water particle acceleration if pipe- line was absent 27mt/T = position of wave cycle over center of pipe section with respect to time kinematic viscosity of fluid mass density of fluid phase shift of maximum lift forces with respect to wave cycle Computer Programs Input Parameters: ANGLE C orientation angle calibration factor for manual digitizer CFD CFU XW YI(1) SYMBOLS AND DEFINITIONS--Continued downward force calibration factor upward force calibration factor bottom clearance downward force calibration factor pipe diameter negative wave (trough) calibration factor wave force readings zero point of wave force record wave surface readings number of wave force readings wave period upward force calibration factor positive wave (crest) calibration factor zero point of wave record length of pipe test section amplification factor for force record amplification factor for wave record wave surface readings Program Variables: orientation angle (in radians) orientation angle (in degrees) horizontal coefficient of drag vertical coefficient of drag bottom clearance CLV CLVA CLVU SYMBOLS AND DEFINITIONS--Continued coefficient of lift (calculated using horizontal velocity in direction of wave advance and projected area in plane parallel to the pipeline axis) coefficient of lift (calculated using horizontal velocity in direction of wave advance and projected area in plane normal to the direction of wave advance) coefficient of lift (calculated using the component of the horizontal velocity in the direction perpendicular to the pipeline axis and the projected area in the plane parallel to the pipeline axis) horizontal coefficient of mass vertical coefficient of mass stillwater depth pipe diameter A we We Umax 1/2 OK We max calculated horizontal wave force measured wave force readings (in grams for two-dimensional data; in 10-grams for three-dimensional data) 2 1/2 p A Ue maximum positive wave force (measured) o V (du/dt) a maximum negative wave force (measured) 0 V (ov/dt) oy measured wave force readings (in pounds) calculated vertical wave force wave height wave surface profile readings PHI CDH CDV CLER CLV CLVA CLVU CMH SYMBOLS AND DEFINITIONS--Continued phase-shift parameter @ of modified lift force equation Tr mass density of water difference between measured wave force and calculated wave force wave force averaged through wave cycle wave period maximum horizontal water particle velocity length of pipe section parameter K of modified lift force equation wavelength vertical distance from bottom to center of pipe section Tabulated Experimental Data orientation angle of pipeline with respect to wave crests horizontal coefficient of drag vertical coefficient of drag bottom clearance coefficient of lift (calculated using horizontal velocity in direction of wave advance and projected area in plane parallel to the pipeline axis) coefficient of lift (calculated using horizontal velocity in direction of wave advance and projected area in the plane normal to the direction of wave advance) coefficient of lift (calculated using the component of the hori zontal velocity in the direction perpendicular to the pipeline axis and the projected area in the plane parallel to the pipeline axis) horizontal coefficient of mass CMV DIA vertical coefficient of mass pipe diameter average horizontal force (averaged over complete wave cycle) wave height parameter k of modified lift force equation wave length phase shift parameter ¢ of modified lift force equation wave period maximum horizontal component of water particle velocity at center of pipe section if absent 4 et oa He FORCES EXERTED BY WAVES ON A PIPELINE AT OR NEAR THE OCEAN BOTTOM by George L. Bowte I. WAVE FORCE ANALYSIS 1. Wave Force Components on Pipelines Near the Bottom. The most common method of analyzing wave forces on pipelines is the application of the Morison equation (Morison, et al., 1950). Using this approach, the total wave-induced force on a pipeline can be broken into several components, depending on whether the components are due to the water particle velocities or accelerations. These force components can, in turn, be separated into horizontal and vertical components by using the horizontal and vertical components of the water particle velocities and accelerations in their respective force equations. Where there is no lift effect and no eddy-induced forces, the vertical component, F\,, of the total wave force is ov hy = Ci, ? Gb. = Gi O Vas? 2 Gy oA viv| (1) and the horizontal component, Fy}, 1s Fh = (Fr), * (Fd), = CM 0 V 32+ 1/2 Cp p A uful, (2) where Ome = vertical component of inertial force (FI), = horizontal component of inertial force Co = vertical component of drag force (Fp), = horizontal component of drag force Vv = vertical component of water particle velocity if pipeline was absent u = horizontal component of the water particle velocity if pipeline was absent OV ; 3 : at = vertical component of water particle acceleration if pipeline was absent 3 - horizontal component of water particle acceleration if pipeline was absent A = projected area of pipe section V = volume of fluid displaced by pipe section 0 = mass density of fluid Cu = coefficient of mass Cp = coefficient of drag For a pipeline located near the ocean bottom, the water particle orbits are flattened parallel to the boundary. Assuming a horizontal bottom, the vertical motions of the water particles are small in com- parison to the horizontal motions, especially in shallow-water depths relative to the wavelength. As a result, the vertical components of the water particle velocities and accelerations are much smaller than the horizontal components, and correspondingly the vertical components of the drag and inertial forces will be smaller than the analogous horizontal forces. Since the water particles at the bottom are effectively oscillating in a horizontal plane, the vertical excursions of the water particles will generally be less than the diameter of a submarine pipeline lying on or near the bottom. Therefore, the vertical drag forces are generally insignificant, and could probably be neglected from the vertical wave force equation. Pipelines near the bottom are subject to vertical lift forces. These forces are the result of the asymmetric distortion of the flow field due to the proximity of the bottom boundary, which induces dif- ferences in the horizontal flow velocities and corresponding pressure distribution over the top and bottom of the pipeline. Since the water particle velocities near the bottom are at a maximum in the horizontal plane, the lift forces induced by these horizontal motions will gener- ally be the predominant force acting in the vertical direction. Transverse "lift" forces due to eddy shedding may also be an important component of the vertical wave force, since these forces are also due to the horizontal water particle velocities and excursions which are maximum in the horizontal direction. Certain values of the Keulegan-Carpenter parameter and Reynolds number must be attained for the eddy release phenomenon to occur. The proximity of the bottom boundary will probably have some effect on the formation and release of the eddies, both because it is a solid boundary, and because it affects the orbital motions of the water particles induced by the wave action. Although the eddy-induced component of the vertical wave force may be significant when compared to the relatively small vertical drag and inertial forces, the experimental results of this investigation show that the eddy-induced lift forces are much smaller than the ''Bernoulli- type" lift forces for pipelines located near the bottom. At large clearances above the bottom where the Bernoulli-type lift effect becomes negligible, the transverse lift forces due to eddy shedding may become a Significant component of the total vertical force. At the same time, as the pipeline is raised farther from the bottom boundary, the verti- cal inertial and drag forces also become more significant. The vertical component of the total wave-induced force acting on a pipeline near the ocean bottom thus consists of four components--the lift force, the inertial force, the drag force, and the transverse lift force due to eddy shedding. Using the Morison approach, the total ver- tical wave force is expressed as the sum of these components: = ! Sa Sie ie, + Fp) 1B (3) where F; is the lift force and Fi is the transverse lift force due to eddy shedding. 2. Wave-Induced Lift Forces. Consider a pipeline in contact with a horizontal rigid, impervious bottom. Water cannot flow between the pipe and the bottom boundary, so the flow must be diverted over the top of the pipe. The asymmetrical distortion of the flow field results in maximum velocities over the top of the pipe section and minimum velocities over the bottom, with zero velocities at the stagnation point on the upstream side of the pipe bottom at the point of contact with the sea floor. Correspondingly, the associated pressure distribution will induce an upward lift force for any velocity field acting on the pipeline. The stagnation pressure at the bottom of the pipe section will increase with increasing veloc- ity, while simultaneously the pressure distribution over the top of the pipeline will decrease with the increased velocities of the flow di- verted over tne top of the pipe section. The wave-induced lift forces will thus act in the upward direction throughout the wave cycle, in- creasing with the horizontal water particle velocities to maximum mag- nitudes under the crests and troughs of the passing waves, and diminishing to zero at the points of horizontal flow reversal. In contrast, a pipeline located at a small clearance above the bottom boundary is subject to a more complex type of lift phenomenon. At the phase in the wave cycle where the horizontal component of the water particle velocity reverses direction, the horizontal velocity over the pipeline is approximately zero. As the wave crest or trough begins to approach the pipeline, the wave-induced horizontal velocities are initially low, inducing unrestricted flow at low velocities over both the top and bottom of the pipeline. However, the water flows 19 faster through the bottom clearance constriction than over the top of the pipeline, so the corresponding differences in the pressure distri- bution exert a downward (negative lift) force toward the bottom bound- anya (halbcyn wl iia)) ee At first, the negative lift force will increase with the increas- ing horizontal water particle velocities of the approaching wave, since the flow velocities increase at a faster rate through the bottom clear- ance constriction than over the top of the pipeline, thus producing larger differences in the corresponding pressure distributions over the top and bottom of the pipe section (Fig. 1, b). This continues until viscous effects begin to restrict the flow through the narrow bottom clearance. For a given small clearance and a given amount of energy in the horizontal water particle velocities approaching the pipeline, the velocities and flow rates of a viscous fluid through the bottom clearance constriction can attain only certain maximum values. Thus, a "choking" effect is exerted on the restricted flow through the small bottom clearance, and the remainder of the wave- induced flow is forced to flow over the top of the pipe section. Cor- respondingly, the stagnation point will shift downward, increasing the pressure on the lower upstream side of the pipeline. The larger the proportion of the flow diverted over the top of the pipe, the lower the stagnation point. At the same time, the increasing velocities associated with the approaching wave crest cause the restricted flow through the bottom clearance to form a turbulent jet with the generation of eddies behind the jet. The generation of increased turbulence and eddies results in an energy loss in the water flowing through the bottom constriction, decreasing the velocities under the pipe section behind the jet. The above effects associated with the choking phenomenon limit the maximum flow velocities and minimum pressures under the bottom side of the pipe section. In contrast, the unrestricted flow velocities over the top of the pipeline increase freely with the increasing horizontal velocities of the advancing wave. The increased part of the approach- ing flow that is diverted over the top of the pipe section due to the shift in stagnation point produces a further increase in the flow velocities over the top. Correspondingly, the pressure distribution over the top side of the pipeline decreases at a faster rate than the associated pressures along the bottom side, so the negative lift force gradually decreases and eventually becomes positive (Fig. 1, c, d, and eve ‘At this stage, the upward lift force becomes larger as the hori- zontal velocities acting on the pipeline increase further with the advancing wave crest or trough (Fig. 1, f). 20 SMALL NEGATIVE (a) fa (ean * oa INCREASING NEGATIVE (b) Lier DECREASING NEGATIVE (c) @ LIFT (d) Via e ZERO LIFT io ee ) fn oy a aa SMALL POSITIVE le (*) LIFT SS (f ) INCREASING POSITIVE ULE Figure 1. Change in lift with increasing velocity. 2 | As the wave crest or trough passes, this series of steps in the lift force phenomenon is reversed. The horizontal velocities approach- ing the pipe section begin to decrease, resulting in a decrease in the positive lift force exerted on the pipeline. As the velocities decrease further with the passing wave, the flow under the pipe section begins to become less restricted. The choking effect thus decreases, and the turbulence and eddies near the bottom clearance gradually diminish. As the flow under the pipe section ceases to be restricted, less of the horizontal flow approaching the pipeline is forced to flow over the top of the pipe, so the stagnation point will accordingly shift upward, closer to the center of the pipe section. The flow velocities decrease simultaneously over the top and bottom of the pipeline as the wave passes, but the rate of decrease is faster over the top of the pipe than in the vicinity of the bottom constric- tion. The positive lift force decreases until eventually, the flow velocities, location of the stagnation point, and associated pressure distribution are such that the pressure integrated over the pipe sec- tion again results in a negative lift force. The downward lift force then increases as the flow through the bottom clearance becomes less restricted with the decreasing velocities of the passing wave. This lift phenomenon, as shown in Figure 2 for a passing wave crest, is repeated twice during each wave cycle as the direction of the wave- induced horizontal velocities reverses under the crests and troughs of the passing waves. In reality, the horizontal flow reversal occurs almost instanta- neously, so the negative lift force does not return to zero at the point of zero velocity when the flow reverses through the bottom clear- ance constriction. The instant of zero velocity occurs only at the center of the pipe cross section (the reference point). Since the pipeline has a finite diameter, the wave-induced flow acting on the pipe section at any instant includes the sum of the flow condi- tions induced by the part of the wave covering the entire diameter of the pipeline. So instead of going to zero with the passing wave crest, and then increasing initially with the approaching trough, the lift force remains negative during the period of minimal velocities as the flow reverses under the pipe section. In a similar manner, the lift force does not become positive as soon as the choking effect occurs in the bottom clearance constriction. The development of the choking phenomenon involves the formation of a turbulent jet through the constriction, and a downward shift in the stagnation point as more water is diverted over the top of the pipe with increasing restriction of the flow through the clearance. The corresponding changes in the velocities, flow pattern, and associated pressure distribution over the top and bottom of the pipe section pro- duce the transition from negative to positive lift. This process requires some small but finite amount of time. Conversely, the reversal 22 *}Sso1D oAeM Sutssed YIM 9dIOF YFTT uT oBuey) 206 —e00L2 | | | | 006 7 | oO oO ‘7 oInsty o0L2 oO a of these processes with the decreasing velocities of the passing wave crest also involves a small but finite amount of time. Thus, there will be a slight timelag in the point of maximum positive lift with reference to the instant of maximum velocity as the wave crest (or trough) passes over the reference point. The smaller the amount of positive lift rela- tive to the amount of negative lift, and the later the positive lift occurs in the wave cycle, the greater the timelag. An example of the lift force phenomenon over a complete wave cycle for a small bottom clearance is shown in Figure 3. For a given pipe diameter and wave condition, as the bottom clearance is increased, higher velocities are necessary to produce the choking effect in which the flow becomes restricted through the bottom clearance constriction. Thus, as the bottom clearance is increased, the flow under the pipeline begins to become restricted closer to the approaching wave crest or trough, where the horizontal velocities are at a maximum; this choking effect also diminishes soon after the wave crest or trough has passed. Therefore, as the bottom clearance is increased, the downward lift force occurs during a larger part of the wave cycle. At the same time, larger clearances permit greater maximum velocities and corresponding lower pressures under the pipe section. Since higher flow rates are possible under the pipe section, less of the wave-induced flow must be diverted over the top of the pipeline. As a result of these changes, the negative lift forces reach a greater magnitude before the choking effect begins, and these maximums are attained later in the Wave cycle. Correspondingly, the upward lift forces occur during a smaller part of the wave cycle, and the maximum magnitude these forces attain decreases with increasing bottom clearance. These maximum values are also reached later in the wave cycle. If the clearance is increased further, a point is eventually reached at which the clearance is large enough so that the choking effect does not occur. At this stage, the velocities are higher through the bottom clearance constriction than over the top of the pipeline during the ~ entire wave cycle. So the associated pressure distribution results in a negative lift force throughout the wave cycle, with maximum downward forces occurring under the crests and troughs of the passing waves. The negative lift diminishes to zero at the points of horizontal flow reversal. As the bottom clearance is increased further, the downward lift effect is gradually reduced. The phase of the force cycle relative to the-wave cycle remains the same, but the magnitude decreases. Eventually, a point is reached where the bottom clearance no longer acts as a constric- tion to the wave-induced flow. The flow pattern becomes approximately symmetrical, and the increased velocities of the horizontal flow diverted Over the top and bottom of the pipeline, along with the corresponding 24 WAVE PROFILE ara eal LIFT FORCE 1go° —<—— —h (a) (b) (c) (d) (e) (£) (g) (h) (i) (3) Figure 3. Lift force phenomenon. Unrestricted flow through the bottom clearance at low velocities results in downward lift force. Unrestricted flow through the bottom clearance at higher velocities increases the negative lift. Writ 5 Choking effect begins, so downward lift force decreases. Velocities increase and pressures decrease at a faster rate over the top of the pipe section than in the restricted flow through the bottom clearance, so the lift force becomes positive. Upward lift force increases with increasing velocities. Positive lift reaches a maximun. Positive lift force decreases and choking effect diminishes with decreasing velocities of the passing wave crest. Lift force again becomes negative as the flow through the bottom clearance becomes less restricted. Unrestricted flow through bottom clearance at low velocities results in downward lift force. Lift force cycle is repeated as the flow reverses with the approaching wave trough. (ao pressure distribution, become approximately equal over both sides of the pipe section. At this point, the lift effect is no longer present, and the lift force term may be neglected in calculating the wave-induced forces acting on the pipeline. The transition in the lift force cycle with increasing bottom clearance is shown in Figure 4. 3. Model for Wave-Induced Lift Forces. The traditional lift force equation, derived for unidirectional steady-flow situations, is expressed as Fy, = 1/2 Ci, p A u*, where Cy, 2s the coefficient of lift. This equation has been applied to wave-induced lift forces, using the horizontal component of the oscillating water particle velocity, u, in the relationship. The lift force expressed in this way assumes that the force acts in one direction only (either upward or downward) throughout the entire wave cycle. A pipeline located on the ocean floor with no clearance will experience an upward lift force throughout the entire wave cycle, increasing with the horizontal velocities to reach maximum values under the crests and troughs of the passing waves, and diminishing to zero as the horizontal velocities go to zero at the point of flow reversal. This phenomena is described adequately by the above lift force equation wikthival pOSi tive! coefrlcient of dake Gye A pipeline located at a large enough clearance above the bottom so that the choking effect does not occur will experience a downward lift force throughout the wave cycle, since the flow is always faster through the bottom constriction than over the top of the pipeline. Again, this negative lift force increases with the horizontal water particle veloci- ties, reaching maximum magnitudes under the crests and troughs of the passing waves, and decreasing to zero as the flow reverses. This phe- nomenon is also suitably expressed by the traditional lift force equation, but using a negative coefficient of lift. These two situations represent the extreme cases bounding the lift force phenomena. However, the choking phenomenon will occur at any clearance between these two limits, and the traditional lift force equation cannot be uSed to accurately describe the forces exerted on a pipeline. This equation must be replaced by a model developed speci- fically for wave-induced lift forces. The experimental results of this investigation demonstrate that the largest wave-induced iift forces occur at these intermediate clearances, where the choking phenomenon does develop. Since the lift force phenomenon is repeated twice per wave cycle with the reversal of the horizontal flow pattern, the lift force can be described mathematically by a sinusoidal function of twice the fre- quency of the waves. In addition, the mathematical expression must allow for description of the following lift force properties: 26 ZO Zn On Ome al OOn mn eOn WAVE PROFILE (a) AL PIPELINE ON BOTTOM O FL (b) SMALL CLEARANCE ) FL (c) INCREASING e) CLEARANCE FL (d) 0) NO CHOKING EFFECT FL (e) O LARGER CLEARANCE Figure 4. Change in lift force record for increasing bottom clearance. eu (a) The lift force may be positive during part of the wave cycle and negative for the rest of the cycle. The proportion of positive lift to negative lift may range from all positive lift to all negative lift. (b) The positions of the maximum values of both the upward and downward lift forces will shift with respect to the position of the wave cycle as the bottom clearance is increased (for a given pipeline and wave condition). (c) As the clearance is increased, the maximum value of the upward lift force will decrease, while correspondingly the maximum value of the downward lift force will increase. (d) When the bottom clearance is increased to a point at which the lift effect is downward throughout the entire wave cycle, further increases in clearance will result in decreases in the maximum magnitude of the downward lift force, but without a shift in the position of the maximum lift force with respect to the position of the wave cycle over the pipeline. A lift force equation of the form, Fy = 1/2 Gy OPA una licoss (0) OP ks (4) allows an adequate mathematical description of all the above properties of the wave-induced lift force phenomena. This equation fits the experi- mental data reasonably well over the wide range of conditions tested. The parameters involved in this modified form of the traditional lift force equation are: Git. =|) (COCHENCHeMt On dake fe) = mass density of fluid A = projected area of pipe section Umax = maximum value of horizontal component of water particle velocity at center of pipe section if pipeline was absent 0 = —— = position of wave cycle over center of pipe section with respect to time, where T is the wave period and t is the time since the last crest passed over the center of the pipe section (see definition sketch in Fig. 5). The wave crest corresponds to @ = 0° (0 radians) or 27t/T = 0 radians. The wave trough corresponds to 180° (m radians) or 27mt/T = 1 radians 28 DIRECTION OF WAVE ADVANCE Figure 5. Definition sketch. — 29 b = phase shift of maximum lift forces with respect to wave cycle k =) negatives traction of elite storcelcyele The parameter, k, represents the increase in the magnitude and duration of the negative lift forces acting on a pipeline with increas- ing bottom clearance, and the corresponding decrease in the magnitude and duration of the positive lift forces. The value of k varies from a minimum of 0 to a maximum value of 1. k = 0 corresponds to the case of a pipeline lying on the bottom with no clearance, in which the lift forces are positive throughout the wave cycle. k increases with increasing bottom clearance to a maximum value of 1, which corresponds to the case of a pipeline located at a sufficient clearance from the bottom so that the choking phenomenon does not occur, and in which the lift forces are therefore negative throughout the wave cycle. The phase shift parameter, », represents the shift in the position of the maximum values of both the positive and negative lift forces with respect to the wave cycle as the bottom clearance increases. The value of $ may range from 0° to a maximum value of 90°. = 0° corre- sponds to the case of a pipeline located on the ocean floor with no bottom clearance, in which the lift forces are positive throughout the wave cycle with maximum forces occurring under the crests and troughs of the passing waves. increases with increasing bottom clearance to a maximum value of 90°, corresponding to a pipeline located above the bottom at a sufficient clearance so that the choking effect does not occur; the lift forces are negative throughout the wave cycle with maxi- mums occurring under the crests and troughs of the waves. As defined, @ = 0° when k = 0, and » = 90° when k = 1, or vice versa. The coefficient of lift, C;, in this form of the lift force equation will always have a positive value, since negative values of the lift force are accounted for by the value of the parameter, k. The lift force equation is shown graphically in Figure 6. To apply the lift force equation to a practical design situation, values of C;, k, and ¢ must be determined for a given set of pipeline and wave conditions corresponding to the particular case under considera- tion. Selection of the appropriate values requires quantitative knowl- edge of the functional relationships between these parameters and the wave conditions, bottom clearance, and pipeline size and configuration. The development of these relationships was the purpose of the experimen- tal part of this investigation. In a real situation, a pipeline on the ocean floor is often laid over an irregular bottom, supported by the high points in the bottom topography but probably spanning the depressed areas. In this case, the pipeline must be broken into component sections of the same approximate bottom clearance for a separate analysis of each section. The results of the 30 “sdozouezed od10F JFTT FO uotiturzog ‘9 oan8ty (2*%°4Ynyd 19 2/1 )(/) sony d 15 2/I (XM NY J. 19 2/1) (1-1) o08l 390Y04 1311 3114d0ud SAVM 3| analysis will yield the lift force record (both magnitudes and time history) of each separate component pipe section, which may then be integrated in the appropriate manner to determine the maximum wave- induced stresses exerted on the pipeline at any critical section. This is important because the maximum lift forces may act upward on a bottom-supported section of a pipeline, while acting downward on the adjacent sections of the pipeline spanning the bottom at a small clearance. Maximum values of both the positive and negative lift forces acting in opposite directions could easily occur at the same point in the wave cycle (under the crests and troughs), thus exerting stresses on the pipeline twice as high as would be calculated consid- ering any pipe section alone, or in using some average clearance for a long section of the pipeline. 4. Extension of Model to Higher Order Theories. The lift force model (eq. 4) is based on linear theory, assuming the lift force phenomenon is identical as either the wave crest or trough passes over the pipeline. Such a symmetrical expression is not flexible enough to consider slightly different kinematics under the wave crests and troughs, which are expressed in higher order theories. These different kinematics would, in reality, produce slightly different lift forces under the crests and troughs of nonlinear waves. The lift force model described above was derived as a modification of the traditional lift force equation using linear wave theory to express the horizontal water particle velocities. Using linear wave theory, the traditional lift force equation can be expressed as: Fra eu/2 Cason alas cos? (6). (5) This equation was modified to make it a suitable expression for wave- induced lift forces by adding the phase shift parameter, >, to account for maximum lift forces occurring in places other than the crest and- trough in the wave cycle, and by adding the parameter, k, to account for positive lift forces during part of the wave cycle and negative forces during the rest of the cycle. This modified equation fits the experimental data very well for all conditions tested in this investi- gation. The model was developed after thorough inspection of the experimental data. For a given pipe diameter and wave condition, the force record followed a sinusoidal relationship of twice the frequency of the waves. As the clearance increased, the maximum positive forces gradually dimin- ished while continuously shifting to a-maximum of 90° from the wave crest as the forces went to zero (Fig. 4). At the same time, the maximum negative forces slowly grew from a minimum value of zero at a position Be 90° from the wave crest and increased while continuously shifting posi- tions to reach a maximum negative value at a position 180° from the wave crest (Fig. 7, a). Since a sinusoidal function of twice the frequency of the wave (sin 20 or cos 20) can be expressed as cos*@, using the appropriate trigo- nometric relationships, and since the lift force is a function of the hori- zontal velocity squared (Umax cos 6), using linear wave theory, the lift force equation was expressed as Fy = 1/2 Cy o A Cae icos? © = @) = eye However, it is the symmetrical properties of this equation and linear wave theory that allow this expression to work so well. When higher order wave theories are applied to this relationship, problems due to nonsymmetry are encountered. This is easily seen by graphically compar- ing the transition from positive to negative lift forces with increasing bottom clearance with this lift model, using both linear and higher order theories. The horizontal component of the water particle velocity for both Stokes' third-order waves and linear waves is shown in Figure 8, along with the corresponding lift forces on a pipeline for the two extreme cases of: (a) a pipeline on the bottom with no clearance, and (b) a pipeline with a large enough bottom clearance so that the choking phe- nomenon does not occur. By gradually shifting the linear theory lift force curve for case (a) (no bottom clearance) to the right 90° from the wave crest, while simultaneously lowering it so that the forces become negative, the lift force curve for case (b) is obtained (com- pare Figs. 7 and 8). This same transformation of the wave force record was observed with increasing bottom clearance in the experimental data. However, if this procedure is repeated with the Stokes' third-order lift force record, the correct force record for case (b) is not obtained (compare Figs. 7 and 8). In reality, rather than a mere shift of the force record downward and to the right with increasing bottom clearance, a simultaneous transformation of the shape of the lift force record would also occur for highly nonlinear waves. This gradual transforma- tion of the shape occurring simultaneously with the shift would provide a continuous change in the lift force record with increasing clearance between the two limiting cases (a) and (b) (Fig. 8). However, the lift force phenomenon is not a direct function of the instantaneous water particle velocity acting at the center of the pipe section if the pipeline was absent. Rather, it is a complicated function of the asymmetrical distorted flow pattern and accelerating velocity field acting on the pipeline, which in turn causes the choking phenomenon to occur, with the resulting change in the relative differences in the flow velocities and corresponding pressure distribution over the top and bottom of the pipeline. Boundary layer flow through the bottom constric- tion, the formation of a turbulent jet and associated eddies, and a cyclic change in the location of the stagnation point with the accelerating velocity field further complicate matters. In addition, the eddies and 33 a. Linear Theory b. Stokes’ Third-Order 0° 90° Paewae 7. IOP POS 30° 0° 90° 180° 270° 360° INCREASING BOTTOM CLEARANCE aver Nan Comparison of linear and Stokes’ third-order theories. Simultaneous shift of lift force record as ¢ increases from 0° to 90° and k increases from 0 to 1 with increasing bottom clearance. 34 *“SOTLOSYI TOpIO-pITy ,SeyoIs pue IesUTT OF Sased oWldTAXS 9DIOF YFTT JO uostszeduojn *g oansty o09E o0L2 o08l 006 00 AsOaYL 19PIO-P4IYL ,SAx01S “q (193443 ININOHD ON HLIM SONVYV3I19) 39404 L4d!1 SAILVOAN (9) (JONVYV3I1ID WOLLOS ON) 39404 1411 JAILISOd ALIQOT3SA 319ILYVd Y3LVM IVLNOZIYOH o09E o0L2 oO08l 006 oO Asoay|, JDaulq “D S5 increased turbulence generated by the jet may be swept back through the bottom constriction as the flow pattern reverses with the passing waves. Because of this, development of an accurate mathematical description of the lift force phenomena for nonlinear waves that would cover the complete transformation of the lift force record with increasing bottom clearance, and yet be flexible enough to allow application of any higher order theory, would be a formidable, if not impossible, task. Since the lift force model developed for linear theory seems to fit the experi- mental data reasonably well, even for waves that were obviously nonlinear, itshould provide a useful tool for engineering calculations, even though it may not be flexible enough and theoretically correct to allow the use of higher order wave theories. The value of the maximum horizontal velocity, Upax, can be calculated under the wave crest using any higher order wave theory; this value can then be used in the linear lift force model, possibly giving a better approximation of the lift forces induced by highly nonlinear waves. II. EXPERIMENTAL INVESTIGATION 1. Experimental Equipment. Model experiments were performed in three different wave tanks. The two-dimensional tests were done in a 1-foot-wide wave channel in the Hydraulic Engineering Laboratory (HEL) at the University of California, Berkeley. The three-dimensional tests were started in the 8-foot-wide Naval Architecture (NA) tow tank, and then continued in the 8-foot-wide HEL wave tank where the majority of the experiments were conducted, both located at the Richmond Field Station of the University of California. The 1-foot wave channel is 100 feet (30.48 meters) long; the 8-foot HEL wave tank and NA tow tank are 180 and 200 feet (54.86 and 60.96 meters) long,respectively. All tests were conducted at approximately the middle of the tanks. A stillwater depth of 2 feet (60.96 centimeters) was used in the two dimensional tests, and a 3-foot (91.44 centimeters) water depth was used in the three-dimensional experiments. A flapper-type generator is located at one end of each of the HEL wave tanks; the NA tow tank has a piston-type wave generator. The wave period is controlled by varying the speed of the electric motors which drive the wave generators. A cam mechanism with a variable stroke length is connected between the drive motor and the flapper, and the wave height is varied by changing the stroke length. A wave filter, consisting of a series of vertical screens, was placed in front of the wave generator in the 1-foot-wide wave channel to smooth out any irregularities in the generated waves due to reflections from the flapper. A permeable beach was installed at the opposite end of each of the tanks to absorb the wave energy and minimize the wave reflections from that end of the wave tank. 36 The wave-induced forces on the model pipe section were measured by a wave force meter designed and built by Al-Kazily (1972). A few modi- fications were made to make the instrument more suitable for this inves- tigation. The same transducer unit was used in all of the experiments, but fittings of different sizes were made to accommodate test cylinders of various diameters. The force transducer consists of a strain bar mounted between two supports. The model pipe section is mounted to the strain bar in such a way that forces on the pipe induce bending stresses on the strain bar. These forces are measured by four strain gages mounted to the strain bar at sections of maximum strain, with two gages in compression and the other two in tension. The strain gages are wired in a Wheatstone bridge, which is connected to a carrier amplifier which amplifies the output from the strain gages. The signal is then recorded on a strip-chart recorder. The original strain gages were Bean-type BAB-13-125DD-120S, and were mounted to the steel strain bar with EPY-150 two-part epox’, and then coated with Dow Corning Silastic RTV silicon rubber for \.aterproofing. Shortly after the beginning of the three-dimensional tests, problems were encountered in the operation of the transducer. These problems were caused by the deterioration of the original strain gage adhesive and coating, so new strain gages were installed on the transducer unit. The new gages were Micromeasurement-type EA-06-125AD-120, bonded to the strain bar with Micromeasurement M-Bond 610 two-part strain gage adhesive, and then coated with Micromeasurement M-Coat D and M-Coat G for water- proofing protection. About halfway through the three-dimensional tests, further problems were encountered in the operation of the transducer unit, probably due to water leakage into the waterproof coating. There was also evidence of corrosion on the steel strain bar, so it was decided to build a new force transducer using a stainless-steel strain bar to minimize corrosion, and encapsulated strain gages to minimize problems with water leakage. The new strain gages were Micromeasurement-type CEA-06-125UW-120. The same strain gage adhesive and waterproof coatings were used, with Micromeasurement M-Coat B along the lead wires to mini- mize the change of water "wicking" along the lead wires to the inside of the coating materials. The transducer mounting arrangement was different for the two- dimensional and three-dimensional experiments. The test cylinder and transducer unit for the two-dimensional tests were mounted between two support brackets on each side of the 1-foot wave channel. For the three-dimensional experiments, the test cylinder and transducer unit were mounted between two long dummy pipe sections, which were in turn mounted to a steel base. The force meter and mounting arrangement is shown in Figures 9 and 10 for the two-dimensional tests, and in Figures 11 to 15 for the three-dimensional tests. A paralle’-wire resistance-type wave gage was used to record the waves passing over the model pipe section. The gage was mounted directly over Sif Support ZZZLZL SPRING STEEL 1/2" x 1716" ee oy DIA. PIVOTED ROD | CYLINDER an ae D. Force Meter PLEXIGLAS ) Support I" x 1/8" Typical 1.0" Figure 9. Force meter and support. 38 ‘ouNTF OY} UT PpoqUNoW TepuT[AD yOUT-sINo0Y "OT omnsty 39 *(JoJOW 9DIOF) UOT}IIAS 4SO, “TT sainsTy (M3SIA dOL) SSINMOIKL QI/ X,.2/1 AVM ,b/I qaals ONIddS aia YSGNITAD SM3Y9S S39VS = -YaLSWVIG,,b/I syqsixaig SV 19IX3 Id 13S NIVYLS OY G3aLOAld ee Gi) "uoTiIsod ut poqjunow uot}DesS SO] “ZT oANSTY (M3IA Salis) eee OE 9I/I —] 6 oS Se NOILD3S NOILD3S Y3L35W S10 NOILOSS Jdld (YSQNIMTAD SHL ONNOYMW LS3L 39YOS SV 191X353 1d Jdid AWWNG SINIOd QUIHL LV) AWWNG SM3YNS L3S € 4| *LIONpsSueL} VILOF pue uot }IeS so] "¢] ernst yA 42 othe S3TONY 1331S ,2 *,2 ON3 0313S *Tepou suttodtd yo stTJeWsYDS "pL orn3Ty MSA 3Q0lS MOIHL 9I/I MOIHL 9171 SU3HSYM SM3Y9S 13S, SUSHSUM- = = ae ana N3 } : eae a3 Was =< = = ——_——— ——— ——— -a3a1vas u3dd01S ls oe Sy eee ee wseeny t= se th H3ddO1S = — — ~~~ ~~ ~~ ~ ~ ~ ~~ ~~~ ~ ~~ = y3eeny (SS3NHOIHL 119M .b/1) ONIGNL SV19IK3Id JA3qNS 00,271 i cl NE a aoy > - Q30VauHL ,8/¢— es SNOILD3S 3did AWWNO — ie -SM3Y9S i SSS OSS SS SS, SS 3 + NOILI3S 1S3L = SE SNOUdV W1L3W 133HS = SA (WO1L10@ 3H SNOW G3073M S3TONV 1331S M3lA dOL ae %,2 OML HLIM HLON31 SNOIV G39804NI3u) 3SV8 1331S ADIL 91/2 43 “[Tepow sutjodtg "ST omnsty 44 the center of the test section, so that the wave records could be cor- related directly with the resulting wave-induced force record. A Brush dual-strain gage amplifier was used in the experiments, with one channel connected to the wave gage, and the other channel connected to the force meter. The amplifier was connected to a Brush two-channel rectilinear writing recorder which continuously recorded the waves and corresponding wave-induced forces on the pipe section (CRieS IS): An electronic digital data acquisition system (Paulling and Sibul, 1968) was used in the three-dimensional experiments. The digitizer was connected in parallel with the strain gage amplifier to record simulta- neously the wave and corresponding force data on magnetic tape, while at the same time the data were being recorded continuously on the strip- chart recorder (Fig. 17). The digitizer sampled alternatingly from both the wave record and force record at a rate of 100 samples per second, resulting in 50 samples per second from each of the two chan- nels. 2. Procedure for Two-Dimensional Experiments. a. Calibration. Both the wave gage and the force transducer were calibrated before each set of experimental runs. The wave gage was calibrated statically by raising and lowering the gage in increments of U.U> TOOT (1.52 centimeters) and recording the output. The force meter was also calibrated statically by hanging weights in increasing equal increments from a system of pulleys connected to the force meter and recording the output on the strip chart. The force transducer was cali- brated in both the upward and downward directions by rearranging the pulley system and repeating the above procedure. The calibration method is shown in Figure 18. b. Procedure. After calibrating the force meter, the model pipe section was lowered and fixed in a horizontal position at the desired clearance above the bottom of the wave channel, with the long axis of the test cylinder parallel to the approaching wave crests. A sliding point gage was mounted to the wave channel above the pipe section and was used to accurately set the model pipe to the desired bottom clear- ance and aline the pipe section parallel to the wave crests. Once the model was in the correct position, the mounting brackets and support struts were clamped to the sides of the wave channel. The force trans- ducer was mounted in such a way that it was sensitive only to forces acting in the vertical direction. After the model pipe section was mounted in position, the wave gage was lined up directly over the center of the pipe section with a plumb bob and then clamped in position. The wave gage was then calibrated as described above. The experimental arrangement is shown in Figure 19. 45 “squounijSsut 3uUTpiodex ysnag “QT omnsTty 46 truments. ing ins Digitizer and record Figure 17 47 “squsWTIedxe [TBUOTSUSWTp-om} TOF poyzoU UOTIeIGTTe) “gt oan3ty NOILVYSITVS GYVMdN A311Nd \ SLYOddNS \ Y3aL3W NOILVYSIIVD GYVMNMOG LYOddNS A341 1Nd Y 48 S359} [BUOTSUSUTP-OM} TOF JUsWesueITe TequoUTIedxg ‘6T oin3sty M3IA LNOYS M3IA 3QGISs aan SLYOddNS 2807 Y3L4W J9YO4 ANVL SAVM J9V9 JAVM =p (op) me [ISS LYOddNS SNILNNOW BWABBUAUAIBBeeeal 49 The pipe model and wave gage were mounted in a glass-walled part of the tank near the middle of the wave channel to facilitate the vis- ual observation of the phenomenon being studied. For each bottom clearance tested, a series of runs was made with waves generated at 19 different wave periods, covering a range of 0.95 to 2.5 seconds. Seven wave heights were generated for each wave period, ranging up to 0.34 foot (10.4 centimeters). After these runs were completed, the pipeline was set at another bottom clearance, and the procedure was repeated. Seven bottom clear- ances were tested for each wave condition, ranging from 0.001 foot, WAN M/S5 S/O, W/4b, Ws euyel Z sinches (O,805, UeS9, S18, 4.76,. aime 6.35 millimeters, 2.54 and 5.08 centimeters), respectively. The mini- mum clearance tested (0.001 foot) was that which placed the pipe sec- tion as close to the bottom as possible without touching the bottom when the waves passed over it. This was necessary to measure any down- ward forces exerted on the pipe section due to the wave action. The 2-inch bottom clearance placed the pipe section far enough from the bottom so that the vertical lift forces were insignificant. These experiments were carried out with a 4-inch-diameter (10.16 centimeters) test cylinder. The experiments were repeated with pipe sections of 2-, 245-, and 3-inch (5.08, 6.35, and 7.62 centimeters) diameters, but only three bottom clearances were tested--0.001 foot, 1/8 inch, and 1/4 inch. The wave conditions covered the same range of wave heights and periods, but were not quite as extensive in number. In addition to the vertical force measurements, a series of experi- ments was performed to measure the horizontal forces acting on the pipe section, so that the resultant wave-induced force could be determined throughout the entire wave cycle for several of the experimental condi- tions tested. Only the 4-inch-diameter test cylinder was used in these experiments, since the corresponding vertical experiments were the most extensive for the 4-inch cylinder. The horizontal forces were measured by rotating the force transducer 90° so that it was sensitive only to forces acting in the horizontal direction. The calibration procedure was the same as described above for the vertical force measurements except that the system of pulleys was rearranged so that the calibration weights exerted forces in the horizontal direction only. All seven of the bottom clearances used in the vertical experiments were also used in the horizontal tests. The wave periods covered the same range as the vertical experiments, but only 6 of the 19 wave peri- OWS Were Wsece—0,95, 25, 1655 1.85, 2.25, eme 2.55 seconds. Mio ort the seven wave heights corresponding to each wave period in the vertical experiments were used in the horizontal tests. The stillwater depth was held constant at a depth of 2 feet through- out the two-dimensional tests. 50 3. Procedure for Three-Dimensional Experiments. a. Calibration. The wave gage and force meter were calibrated before each set of experimental runs. The wave gage was calibrated in the same manner as the two-dimensional tests, but 0.1-foot (3.05 centi- meters) increments were used rather than 0.05-foot increments, since larger waves were used in these experiments. The force transducer was calibrated in the upward direction in the same manner as the two-dimensional tests, by hanging weights over a pulley to a string attached to the pipe test section. However, because the three-dimensional model was mounted to a base with a small bottom clearance, it was impossible to calibrate the transducer in the down- ward direction by using a system of pulleys, since there was no room for a pulley between the pipe section and the base to which it was mounted. Rather, the force meter was calibrated in the downward direc- tion by placing the weights directly on top of the center of the sub- merged test section and using the submerged weight of the weights in calculating the calibration curve. Weight increments of 50 grams were used in calibrating the transducer. The calibration method is shown in Figure 20. b. Procedure. An overhead crane was used to lower the pipeline model and base into the wave tank. The assembly was first submerged to a depth of about 1 feet (45.7 centimeters). The model was tilted at both ends to remove all air bubbles from the system, and the ends of the dummy pipe sections were stoppered to prevent waterflow through the pipeline model. The bottom clearance between the base and the pipe model was adjusted by placing spacers on the support rods between the base and the dummy pipe sections, and then tightening the nuts on the support rods above the dummy pipe sections. The test section was tnen centered and adjusted carefully to the exact bottom clearance desired with the aid of 10 adjusting screws. The calibration string was attached to the test section, and the assembly was lowered to the bottom of the tank. The calibration string and pulley system was alined directly over the center of the test section with a plumb bob, and the pulley sup- ports were then clamped to the sides of the wave tank. The transducer was first calibrated in the upward direction, after which the calibra- tion string was removed, and the transducer was calibrated in the down- ward direction, as described above. The pipeline model was positioned at the desired angle of orienta- tion on the tank bottom by lining up one of the long edges of the model base parallel to the correct line marked on the bottom of the wave tank. Lines were marked on the tank bottom in 15° increments from 0° to 75°, where 0° corresponds to a pipeline parallel to the approaching wave crests. After the model was calibrated and placed in position, the wave gage was lined up directly over the center of the test section with S| "squowtredxe [PBUOTSUSWIP-S991Y} TOF poyyow uoTzeaqt{e) ‘oz oan3Ty NOILVYSIIVD GYVMNMOG NOILVYSI1VS GYvVMdN ge ee Scere eee 6 Te eee “aSV€ ONIYLS LHSISM NOILVYEIIVO a (ANVL SAVM Ol Q31LNNOW) 4 LYOddNs A3d11Nd A311Nd DZ a plumb bob, clamped in position, and then calibrated as described above. The experimental arrangement is shown in Figure 21. For each bottom clearance, six angles of orientation (0°, 15°, 30°, 45°, 60°, and 75°) were tested. Fifteen runs with different wave con- ditions were made for each bottom clearance and orientation angle. These runs covered four wave periods ranging from 1.4 to 2.6 seconds, with waves generated at four heights for each period, ranging to a maxi- mum of about 0.7 foot (21.3 centimeters). Eight bottom clearances were tested, ranging from 0.001 foot, 1/16 inch, 1/8 inch, 3/16 inch, 1/4 imeh,, 1/2 anch, 1 anch,* and 2 inches. The above experiments were done using a 3-inch-diameter pipeline model. The tests were then repeated using a 2- and 4-inch-diameter pipeline. The 1- and 2-inch clearances were not tested because the lift forces at these clearances proved insignificant in the previous tests. Also, the tests at an orientation angle of 75° were eliminated, since the previous experiments demonstrated that the vertical forces measured at this angle were insignificant, and too small to be measured with any accuracy. Aside from these changes, the 4-inch-diameter pipeline was tested at the same bottom clearances, orientation angles, and wave con- ditions as the 3-inch-diameter model. The 2-inch-diameter model was tested at the same bottom clearances and wave conditions, but only three of the five orientation angles (0°, 30°, and 60°) were tested. The stillwater depth in the wave tank was held constant at 3 feet throughout the three-dimensional experiments, but since the base of the pipeline model was located 2-7/16 inches (6.19 centimeters) above the tank bottom, the effective stillwater depth over the pipeline base was 2.797 feet (85.25 centimeters). The definition sketch for the three- dimensional experiments is shown in Figure 22. 4. Data Reduction. The wave force data were taken on a two-channel strip-chart recorder with the paper advancing at a speed of 25 centimeters per second. One channel recorded the forces while the other channel simultaneously re- corded the wave surface profile directly over the center of the pipeline test section, thus allowing direct correlation of the two records. The two-dimensional experimental data were digitized manually using a Gerber digital data reduction system connected with a card punch to automatically punch the digitized values on computer cards. Using a variable linear scale, each force record was first divided into 20 equally spaced intervals per wave, each interval representing a time ‘interval of T/20, where T is the wave period. Each force record was digitized at these points over an interval of two consecutive waves (beginning at the wave crest), thus giving 40 values for the analysis and averaging the wave forces over two wave cycles. 53 *S4Soq [TBUOTSUOUTp-901Y} TOF JuoWeSueTIe [equowtsredxg “TZ sIN3TY M3IA LNOYS ONILNNOW SLYOddNS ddid AWWNG M3lA 3QIs 54 DIRECTION OF WAVE ADVANCE Diabet nee BOTTOM CLEARANCE mi Figure 22. Definition sketch for three-dimensional experiments. 55 The points in the force records corresponding to the wave crests were chosen as the origin (and end) of the digitized records. These points were determined by averaging the midpoints of three or four hori- zontal lines drawn through the crests of the wave record at several elevations above the stillwater level (SWL). These midpoints were approximately identical except for some of the larger, longer waves in which the peak of the wave crest did not exactly coincide with the mid- point of the zero crossings of the wave crest. A sample data record is given in Figure 23. The three-dimensional experimental data were handled differently; the data were recorded on magnetic tape with an electronic digital data acquisition system. This instrument sampled alternatingly from the two channels (wave and force) at a rate of 100 samples per second, result- ing in 50 samples per second from each channel. The origin at the wave crest and the wave period were determined from the digitized wave records, rather than directly from the strip- chart records. Since positive readings of the wave profile corres- ponded to the crest and negative readings corresponded to the trough, the point of origin of the wave crest was determined by taking the mid- point of the positive readings between zero crossings on the wave profile. The crest was thus defined as the data point closest to the midpoint of the zero crossings. The wave period was determined from the number of readings between two successive crests, since there was a time interval of 1/50 second between each reading. Thus, the wave period was determined to the nearest 0.02 second. The origin of the force record was taken as the force reading cor- responding to the defined origin at the center of the wave crest surface profile. In reality, there was a small timelag of 1/100 second between the wave profile readings and the corresponding force readings. This small timelag was ignored in the analysis, since it was felt that the accuracy of the defined origin at the wave crest was only good to tne nearest 1/50 second, the time interval between successive readings of the wave record. Only one wave cycle was used for analysis of the electronically digitized data. Since the data were on magnetic tape, it was impossible to determine that two successive waves had exactly the same period and height until after the calculations were completed on the computer. Thus, if the waves had slightly different periods, the time phase cor- relation of the corresponding force readings would be slightly in error when taken over two wave cycles. In addition, since the accuracy, resolution, and rapid sampling rate of the electronic digitizer allowed more readings per wave cycle than the manual digitizing method, a suf- ficiently large number of force readings could be obtained in one wave cycle. 56 ‘paodser ejep Jo oftdwexg ‘“¢z omn3Ty i {—+ | + + +} —+_} t t—+, +—+—+ . : 2S 72 SSS SSS SS SS a= ===] === == = SS SS iF SSSaao == 2 =! = == === ======== = SS SS SS SS SS se S25 == a5 2 2=====S====2==2 SS == SS = 22================= === ======= === s========== ====2 ===> S— ==> SSS SSE 4 ====———=— = === ===> 2 = == == === : = Fs a= 7 == === =: Be 22= SSS = 2a == = + 44 4 oe ee SSS 223>>F2= SF: = ===> >= 2 ===> SS ===> === = 7 ee ae SS ==: = = SS 22S SS SSS SSS= ===> SSS SS SSZZZ= BSBEa=Z=z= OG An estimation of the accuracy of the experimental measurements, along with the sources of error, 1s tabulated in the Table. A least squares analysis was performed on the digitized force data to calculate the parameters, C,, 9, k, Cy, and Cp, of the vertical wave force equation, and the coefficients, Cy and Cp, corresponding to the horizontal wave force equation. Using this approach, values of the wave force parameters that best fit the force data throughout the entire wave cycle can be determined. These values were then substituted back into the wave force equation to calculate the force over a complete wave cycle, thus allowing comparison of the results with the original data. The least squares analysis is given in Appendix A. The computer programs used for the analysis are given in Appendixes B, C, and D; the tabulated results of the analysis are in Appendixes E, F, and G. Examples of the computer output showing comparison of the original data with the forces calculated using the results of the least squares analysis are given in Figures 24 and 25. III. RESULTS AND DISCUSSION 1. Resultant Force Through Wave Cycle. Both horizontal and vertical force measurements were made for some test conditions in the two-dimensional experiments using the 4-inch- diameter cylinder. The resultant force throughout the wave cycle could thus be determined for these conditions. Figures 26 to 32 show the resultant force plotted for each bottom clearance under the same wave condition, a period of 1.85 to 1.86 seconds and a wave height of 0.24 to 0.25 foot (7.32 to 7.62 centimeters). Values from the corresponding horizontal and vertical force records were plotted at 20 evenly spaced intervals (18°) through each wave cycle. The forces were plotted for two consecutive wave cycles to indicate the degree of scatter in the data. A rectangle was drawn at each plotted point to illustrate the horizontal and vertical range of the force data over the two wave cycles, and an envelope curve was drawn over these points. Examination of these plots as a group (Fig. 33) shows the transition of the resultant wave-induced force with increasing clearance for the given wave condition (T = 1.85 to 1.86 seconds, H = 0.24 to 0.25 foot). The vertical component of the wave force is dominated by the lift force, while the horizontal component of the resultant force is due to the iner- tial and drag forces, with the inertial forces predominating for the experimental conditions tested. For the smallest clearance (0.001 foot), in which the pipeline is almost in contact with the bottom, the resultant force attains a maximum upward value under the crests and troughs of the passing waves. The total wave force acts in the upward (positive) direction throughout the complete wave cycle, except for small downward forces in the vicinity of 90° and 270°, where the horizontal flow reverses. 58 UOTIEIQT[TeO 9d10F 03 yoO0dsox y.IM ToTFTTdwe Fo ATT TqQeAs pojunow sem 4T YOTYM 07 OSeq dU} FO o8po 9yi YITM Topow suttedtd JO AJUOWOUTTe pue ‘wWoIIOq YyUeI 9Yy2 UO SopTsue oy. SuTyIeW sauT{T Fo Adeandoy S}S9} [BUOTSUOUWTp-99TYy} UT 9sdURIeETO yas 02 posn soses3 ~Teqou fo AdeINdOe $S54S9} [TBUOTSUOUTPp-OM} UT ddUeTeaTO yas 02 posn o3e3 yutod Fo yuMODd 4seeqT pe yonz3suU0d dI9M STOPOW oY} YOTYM WorF Butqnyz JO SIOJOWeTP TBUTWOU UT sUOTIeTIeA JUSWSINSePOW JOOLTG Sqjuowtzedxe [BVUOTSUSUTP-991Y} IOF PIODST oAeM FO SSUTPPOL IOZTIISTp oATSS9DNNS useMIEqG TBAIOUT out} fsquewtredxe [euoTSUSWTp -OM} IOF spLOdeXL yreyo-dt41zs Fo AdeINddy uoT}eIQTTed 03 yOedsoxz YIM TOTFZITdWe Fo AATTTqGeIS JOIIS FO 9DINOS JOLEW quoozed QT UTYIIM 02 93eINDDe Oe YOTYM *S4S9} [BUOTSUSUTPp-90tYy. OY. UT (,SZ pue ,09) seTsue uotT,eJUsTIO ysSedIeT qe usye} eiep rox ydeodxe ‘yudo0zed ¢ os Il $259} [TeuUOTSUOUTp -991Y4} LOZ (LOVOWTTTTW ZST°O) 200F S000°0 £S}S9} [BUOTSUSUTp-OM} LOF OOF T[OO'O (1oZOWeTp uo BSutpusdep ‘jueored 7°] 01 9°0) (AO JOUTTTIW OT9°0) 300F Z00°0 (s3s02 TVUOTSUSUTp-901Yy} IOF JusdZed s¢*yg ‘S3450} [VUOTSUSUTp-OmM} TOF YUedTEd ¢s*Q) YOUT g/T (potzed uo 3ut -puodep ‘jusor1ed pT 02 9°Q) spuodses 70°0 qusszed s 01 ¢ IOLILG WNuUTXeW ODLOF OACM eTsue uoT}e4UNTIO aouereeto wlo.70g Zojouwetp odtg yqjdep 1917eM potzed oaeM 2YySTOY ARM aT qe tse) “S]ZUOWOINSeoU [eJUoWTIedxs Fo ADeINdde pszeUt sy “oT qeL 59 CALDATA TOTAL ee77 Cj * ~FP(LB) 022571 -e16C61 012541 —eC7806 -2C6428 —eCSCS1 204592 —0C6734 -0C 9642 =214387 019132 226969 -ez1581 219285 -214cal 208112 200155 208517 215330 022143 226324 028492 e29111 225705 220750 0139 36 °C 4645 -0€ $969 017448 =026939 233060 037346 -23e876 =o 36427 233519 027091 Figure 24. KOATSezS suM sa 3276 civ 2479 + FV(LA) 224647 017757 -21168¢ 2070858 004419 203824 005155 267994 011703 015523 018673 020452 220348 018092 -013719 2007559 - 200208 oO 7EO3 014828 021270 226254 229065 ©2921 226479 °20982 013136 93627 296669 -016785 - 025772 232808 =o3% 2H 238903 037647 - 233835 228048 Example of computer output for vertical least 05720776 HTCFT) of 53 COS24 2COK748 AY 2615694 CLVA 62326 RES(LB) 001977 = 200304 - 200871 - 200718 -202C09 201226 2C0S64 201259 2C206C 201136 200459 -200515 001233 201193 000362 200553 200363 200974 200506 200874 200c7c -200573 -.00098 -200775 200232 200800 2001018 200700 -2000€63 = 001166 2000252 “= 2 S0658 200027 eO1226 200316 2000955 21680 el Sel IMAY7H WAVEL(FT) Ge7d DEP (FT) 20797 SIN24 12399930 COSA 10 26334C AX 2188645 AQ 0198582 Fuv 0072717 Fay © C38916 civu 7.305 cu 4911 FAVG(LA) 70073521 FMAX(LB) e2oill oe oe os oe measured force calculated force measured force - wave surface 60 JMAX(FPS) SEQe00C37 2 SINA/SINAS 2100544 AZ 2961452 FOV 2000929 cov 660182 FMIN(LAD - 038376 Qe ees oe oo CLEARQ(FT) t°) OIACFT) CYL LGTH(FT) 0333 0917 VARIANCE ef 05718 vA 0129201 PHI K 26945 42074 ° oe ° +e calculated force squares ANG( DEG) 3900 analysis. * FP(LaD 015177 201982 -ez2aag 0 25596 051138 —0£9843 —0€£233 -2€3107 081138 26554 014145 2092886 _eziei7 074148 046475 o52171 _ef€912 of3119 043620 e21702 014223 -00Z176 220673 aele ~e St 7RE ~eSS441 =055803 ~eff579 ~2%€766 274818 21632) 20 I2646 017074 Figure 25. * = measured force + = calculated force SEC) HTCFT) waVEL(FT) DEPCFT) JMAX(FPS) CLEAQ(FT) OLACFT) = CYL LGTHIFT) 33 <= 12.A> 223900 0359 e019 0333 e917 TOTAL SuM SQ SIN cosscoss 60679549 62 2€4509 0479832 ap az 0550459 el 7585S Fay FOH 2190450 2038005 rT) con 20991 0706 Favc(Le) FMAX(LA) FMIN(LBD 2924153 S691 265283 + ° — <_———_ — FH(LB) PES(LA)D 0 ue o17a8¢ =00270S el ae 200339 -.90746 el 220661 202198 ee 6 2 038270 202468 es Te °50063 -2C0475 + el 055066 204777 e o ef4075 =e 1taca e ° 1 250729 012376 * ° Lf (-e44G72 -.07ceF es + ° I —el2'04 —207800 ee ont 7017385 203741 cad le 200839 2 02CC? te _ 020661 eOLlse ° ° 039379 9 =-.04227 . I a ©5066? —2041es omit a6 2055064 ~eC2b4F ot ee 984979 ecetzs Ie e 050729 29239) le oe 044073 == 200425 el ee 033190 sedlest ol ae ol 7ABS ~eC MET e I ee 090339 8 -201237 Sol 02061 -eC00l2 * ol 238379 OPRRER oe le 050562 297877 a8 le -05506€ -eC0a2e . el —0£4079 =-.05764 . oO. -e50729 -.CsPec e . I 7044077 7092732 *\¢) el ~e%3194 8 -.21624 oe ol 917495 eOMSES . 5 eCO334 2027C7 te 220868 oat ee 238370 ° 1 * . 250557 ° I s . oosCer . I . e Onde) el ee 059729 ol ae 949973 . I * 6 2372196 =e 0foKHr et sae * = measured force - calculated force Example of computer output for horizontal least squares 6| ANG( DEG) Oo analysis. 2477777 “JysToy YOOF-y7"Q pue ‘potied puodas-cg-{ ‘Q0UeIBSTI 1OOF-T00°O OF aTOAD oAeM YBnoIYI 9d10F JUeITNSOY See =~ eS "9Z osnsTy 62 "3y3ToYy .0OF-p~z°O pue ‘potazed puosds-9gg"T ‘99ueIedTD YOUT-9[/T OF 9TIAD oAeM YBNoLYI 9dTOF JuBA[NSOY “LZ o1n3 Ty y Z0- €0- ¥0- og 63 "1YyStey 100F-S7°Q pue ‘potted puodes-Gg" ‘Q0URLBSTI YOUT-8/T TOF 9TOAD oACM YSsnOIY. sdIOF YUeRTNSOY "gz oan3Ty "2yB3tey 100F-Gz"°Q pue ‘potzed PuodeSs-Sg°T “Q0UeIv9TD YDUT-9T/¢ LOF OTOAD OAEM ysnory} sd10JF JueI[Nsoy "67 9an3B Ty 65 “1y8Tey 10OF-Sz°0 pue “potied puodas-9Qg°T ‘g0ueZeoTO YOUT-p/T TOF eTOAD sAEM YBnory sdLOF UB TNSOY "Og eansTy 66 “yy3toy 100F-~7°0 pue ‘potsed puosss-98"T ‘g9uRIeoTD YOUT-T IoZ oTIAD OAEM YBNOIYZ 9dLOF JuvIT[NSOY "Tg eins ty 67 *1Yy8ToeYy 100F-Sz7°Q pue ‘potszed puodds-9g "J ‘90ueLTeoTD UYoUT-7z TOF aTOAD aAeM YSNoIY} 9d10F JueL[NSOY "7g einsTty 68 eo Fy 0° Clearance 1g0 0.00! ft @ Fh 9 270° * in | B in ay 90° i 270° ~ in ! in 2 in >2 in Figure 33. Change in resultant force with increasing clearance. 69 As the clearance is increased to 1/16 inch, the maximum upward forces decrease in magnitude, and also occur slightly later in the wave cycle. At the same time, the downward forces increase, reaching their maximum values at approximately 90° and 270°. Further increases in the bottom clearance produce a continuous shift of the positions of both the maximum upward and maximum downward forces later in the wave cycle. Simultaneously, the forces become downward rather than upward for a larger part of the cycle. At the same time, the vertical components of the wave force under the crests and troughs become negative and increase in the downward direction, while the negative forces at 90° and 270° gradually decrease to zero. At a l-inch clearance, the resultant force acts downward throughout almost the complete wave cycle, with maximum downward forces occurring under the crests and troughs of the passing waves. The vertical forces are zero at 90° and 270°, the positions of the maximum horizontal iner- tial forces. However, the lift effect is not very large for the 1l-inch clearance. The resultant force plot for the 2-inch clearance shows that the lift effect is still present, but is relatively small, even in com- parison to the small vertical inertial forces. At a slightly larger clearance, the lift effect will disappear, and the vertical forces will be due almost entirely to the inertial forces, since the vertical drag forces are negligible near the bottom. At this clearance, the inertial force will act upward under the trough and down- ward under the crest, so the resultant force plot will take the form of an approximately symmetrical ellipse. This condition is shown in Figure 34 for a smaller wave period (1.23 seconds), with a l-inch bottom clearance. The ellipse is distorted slightly, due to the small drag forces acting in the horizontal direction, 90° out of phase with the larger horizontal inertial forces. The horizontal components of the resultant wave force are also affected by the proximity of the bottom boundary. Although the horizon- tal water particle velocities and accelerations increase with distance above the bottom, the corresponding horizontal drag and inertial forces are larger when the pipe is close to the bottom than when it is located above at larger clearances. Figures 35, 36, and 37 show the resultant force plots at both large and small bottom clearances, for a wave with a period of 0.95 to 0.96 second and a height of 0.24 to 0.25 foot. Because the wave period is small, the horizontal excursions of the water particles at the bottom and the duration of the horizontal flow are too small for the lift effect to develop. So the forces acting in both the horizontal and ver- tical directions are mostly inertial, with a small drag component in the horizontal direction. The resultant force plots therefore take the form of an ellipse. é 70 Saoueles[ YOUT *2YySToy JOOT-¢°Q pue ‘potazod puodes-¢Z'T JT ToZ oTOAD OAM YSNOTYI 9dLOF JUeA[NSOY “pe oinsty ral “qy8tTey .00F-p~7Z°O pue ‘potzed puodses-G6'0 ‘90UeIBOTD 1OOF-100'°O OF S9TOAD 9AeM YBnoOIY} sdLOF JULI TNSOY "gg orn3Ty ee "1y3ToY 00F-p7°0 pue ‘potasod puosas-c¢g"Q “aueIeaTD YOUT-9T/T TOF aTOAD oAeM YBnoIY. 9dA05 quez[NSoy lO ybnosy 20 ( qt) 49 "9g Omn3Ty as “IYySToy 300F-¢7°9 pue ‘pottod puosas-9g6'Q “OOUPIVSTI YOUT-Z LOF OTIAD oACM YSnoLYA 9dI0F JUeI[NSOY "LE eInsTy 74 However, the horizontal components of the resultant forces are larger at the smallest bottom clearances, even though the lift phenomenon is absent. The presence of the bottom boundary produces an asymmetric flow field around the pipeline. The resulting velocities and accelerations of the water particles over the pipe section are thus modified by the presence of the boundary, and the associated horizontal forces are larger than they would be if subject to the same kinematics in the absence of the boundary. The increased horizontal forces on pipelines located close to the bottom are reflected in increased values of the coefficients of mass and drag, Cy and Cp. 2. Orientation Angle Considerations. The coefficient of lift calculated in the least squares analysis of the experimental data was computed using two alternative approaches (Fig. 38): (a) the total horizontal water particle velocity in the direction of wave advance, with the projected area of the pipeline in the plane perpendicular to the direction of wave advance; and (b) only the component of the horizontal water particle velocity perpendicular to the pipeline axis, with the projected area in the plane parallel to the pipe- line axis. After tabulating the data from the three-dimensional experiments, it became apparent that the second method gave consistent values of the coefficient of lift for all angles of orientation. In contrast, the values of Cy; obtained using the first method gave values that were low, and which decreased with increasing angles of orientation (where 0° corresponds to a pipeline parallel to the wave crests). Relationships between the coefficient of lift, Cj, and the parameters, g and k, of the lift force equation were the same for all angles of orien- tation when C;, was calculated considering only the component of the hori- zontal velocities perpendicular to the pipeline axis. In addition, relationships involving any of the parameters of the lift force equation (C;, >, or k) and various dimensionless parameters defining the wave and pipeline conditions were consistent for all angles of orien- tation when the horizontal water particle velocity acting on the pipe section was treated by considering only the component perpendicular to the pipeline, and completely ignoring the parallel component. Thus, the results of this investigation show that the modified lift force equations presented in this report can be applied to pipelines located at any angle of orientation with respect to the wave crests. However, only the component of the horizontal water particle velocity perpendicular to the pipeline axis should be considered as contributing to the wave-induced lift force acting on the pipeline. Using this approach, the parameters, Cy, >, and k, defining the lift forces exhibit the same quantitative relationships between the various dimensionless parameters de- fining the wave and pipe conditions, regardless of the angle of orientation. US DIRECTION OF WAVE ADVANCE Figure 38. Alternative approaches for handling pipeline orientation angles. 76 3. Interrelationships Between Cy, , and k. » and k were defined as varying from 0° to 90° and 0 to 1, respec- tively, with increasing clearance. @ = 0° and k = 0 correspond to the case of a pipeline in contact witn the bottom (no clearance), while the maximum values of @ = 90° and k = 1 correspond to the case of a large enough clearance so that the choking phenomenon does not occur at any time throughout the wave cycle. Since a simultaneous increase of both parameters was noted in the data for increasing clearance between the two limiting cases, it was suspected that a direct relationship may exist between 9 and k. Such a relationship was found, as shown in Figure 39. The same rela- tionship held for all three pipe diameters tested, regardless of the ori- entation angle, indicating that the relationship was independent of these two factors, and was thus valid for any pipeline configuration in which the lift effect was present. In this plot and the ones that follow, the data for orientation angles from 0° to 30° were plotted for each pipe diameter, without differentiating the data corresponding to each angle. The relationships shown were found to be valid regardless of the angle of orientation, provided the data were handled as discussed above (using the component of the horizontal velocity perpendicular to the pipeline axis). The data corresponding to each pipe diameter are distinguished by using different plot symbols. The same relationships hold for orientation angles of 45°, but these data were not plotted in order to minimize scatter so that differences between the pipe diameters could be detected more easily. In general, the same relation- ships held for orientation angles up to 60°. But in some cases, the lift effect was negligible at high orientation angles, so the values of the associated parameters (Cy, ¢, and k) were less accurate. Thus, plotting all of the data corresponding to the larger orientation angles would intro- duce additional scatter, obscuring the valid relationships which were consistent when the lift forces were significant. A relationship was found between the coefficient of lift, Cy, and the parameters, ¢ and k (Figs. 40 and 41). Cy; appears to be better correlated with k than with ». Note that for minimum values of k and 9, corresponding to the case of a pipeline in contact with the bottom, the value of Cj, is approximately 4.5. This value is of interest, since it agrees with the potential flow solution (Cj = 4.495) for the value of the coefficient of lift for a circular cylinder in contact with a plane wall, subject to an inviscid steady flow (Yamamoto, Nath, and Slotta, 1973). Maximum values of Cj occur at approximately k = 1/2, corresponding to maximum lift forces that are equal in both the upward and downward direc- tions. The average value of the coefficient of lift at this point is about 9.0, with values extending up to about 10.5. These maximum values of Cy are attained at approximately » = 25° to 30° in the $ versus Cy; plot. Since the coefficient of lift, Cj, defines the combined magnitude of both the positive and negative lifts, it can be separated into two parts: (a) the part defining the magnitude of the positive lift, C,(1-k), and (a oot vid vid ¥sLIWVIA 08 WI-t NI-€ WI-2 x Vv o "y SNSLOA > €°930) 0+ "6g 9In3Ty IHd 4 78 2° | SNSIOA T 2) x ‘Op omn3TY vid via y3JiINVIA HI—-6 W#I-€ WI-2 m9 oot *d SNSIOA 1, €°930) Ov ‘Tp ean3ty IHd 02 vida VIO YILINVIO NI WI-€ WI-2 Xx Vv © 80 (b) the part defining the magnitude of the negative lift, C;(k). The quantities, C; (1-k) and C;(k), can be referred to as the effective posi- tive coefficient of lift and the effective negative coefficient of lift, respectively. Since Cy = 9.0 for k = 1/2, both C;(1-k) and C;(k) are equal to 4.5 at this point. This means that the lift forces can reach the same maximum magnitude in both the upward and downward directions as are attained in the upward direction only for the same pipe in contact with the bottom (where C,; (1-k) = 4.5, but C;(k) = 0). The effective positive and negative coefficients of lift are plotted versus both $¢ and k in Figures 42 to 45. Again, the correlations are much better with k than with >. The average value of C,(1-k) drops only slightly between k = 0 and k = 1/2, but for values of k greater than 1/2, the effective positive coefficient of lift drops rapidly to a value of O when k = 1. The average value of C,(k) increases with k until it reaches a maximum value of about 6.0 when k = 0.75, and then decreases to about 4.5 when k = 1. Individual maximum values of C;(k) attain values slightly greater than 7.0 in the vicinity of k = 0.75. But even the average maximum value of 6.0 for the effective negative coefficient of lift indicates that the downward lift forces may attain maximum values 33 percent greater than the maximum possible lift forces acting in the upward direction. Maximum values of C, (k) corresponds to a value of » of about 45°, which is half way through the phase shift cycle. The potential flow theory gives a value of C,; = 4.495 for zero bottom clearance, with a discontinuous jump to very high negative values of Cy, for a very small clearance (Yamamoto, Nath, and Slotta, 1973). In the potential flow solution, the value of C, depends only on the relative clearance; i.e., the ratio clearance-diameter. The coefficient of lift is negative whenever the pipe is not in contact with the bottom, and its magnitude decreases as the relative clearance is increased. Although the potential flow solution appears to work reasonably well when a pipeline is touching the bottom, this approach does not work when there is a small clearance. This is because viscous effects are very important for the flow through the narrow bottom clearance constriction. The choking phenomenon limits the maximum flow velocities and corresponding pressure drops on the bottom side of the pipeline, thereby limiting the maximum possible downward lift forces. The results of this investigation indicate that the effective negative coefficient of lift, C,;(k), can attain a maximum value Or only 7oO, Ws is much less than the values of C; suggested for small relative clearances by potential flow theory. The coefficient of lift is obviously not a simple function of relative clearance, since for a given clearance and diameter, both the lift effect and the coefficient of lift will vary with the wave- induced flow conditions. For the smallest relative clearances, the positive lift forces were larger than the negative lift forces, especially where the horizontal water particle velocities and excursions were high. 8 | OIAMETER DIA OIA 2-18 3-18 4-1 © 4 x Cw-TIE193 82 Effective positive coefficient of lift versus k. Figure 42. vio vio y3JLINVIO *) SNSIOA YFTT FO PUSTITFFOOS dAT4TSOd 9ATIIOFFA c°930) Lud 08 09 0% 02 WI-6 X wi-e V WI-2 © ‘op oun ta 02- 0 CA-TIeT19 83 "y SMSIOA YFT[T FO JUSTITFFOOD OATIEZOU OATIIOFFY ‘yy eaIn8Ty vid via Yslawvida WI-» WI-¢€ WI-2 x< Vv i+ We1I 84 oot *$ SNSIOA YFTT FO JUOTOTFFOOD OATIESOU 9ATIIOFFA "SV oins ty €°930) Id 0 Vig WI-6 vida WI-€ YaLINVIGO WI-2 Oo QXx Ne@19 85 The largest negative lift forces do not occur at clearances where the choking effect is absent (corresponding to k = 1 and > = 90°). Rather, the largest values of the effective negative coefficient of lift correspond to values of ¢ = 45° and k = 0.75. Interestingly, when k = 1 and $ = 90° where the positive lift forces have decreased to zero and the choking effect does not develop, the maximum effective negative coefficient of lift is approximately 4.5, the same magnitude as the potential flow solu- tion for the positive coefficient of lift for zero bottom clearance. However, as the bottom clearance is increased further, k and $ remain at 1 and 90°, respectively, while the effective negative coefficient of lift decreases to zero (with the diminishing lift forces). The sionificanmce of these results is easily seen by following these relationships for a given pipe and wave as the pipeline is raised from the bottom, and kigoes fromi0 toll) ineehe unterval fom) , k, Cy, Cj,,(1-k), and Cy.(k) were the same for all pipe diameters tested, regardless of the angle of orientation (provided that C; was calculated considering only the com- ponent of the horizontal velocity perpendicular to the pipeline axis). Thus, for the range of conditions tested, these interrelationships were independent of the scale and configuration of the pipeline. Also, there is no mention of the wave conditions, which indicates the interrelation- ships are independent of the wave conditions as well. The relationships between the parameters, C,, o, and k, defining the lift force equation are useful, since if either ¢ or k is known, the other two parameters can be determined. All that is needed is a rela- tionship between # or k and the wave and pipeline conditions. 86 There appears to be a better correlation between k and the parameters involving C, (CL, C, (1-k), and Cy (k)) than between the analogous relation- ships using 9», so the former relationships should be used. Also, in com- paring the plots of Cj (1-k) versus k (Fig. 42) and C, (k) versus k (Fig. 44), the scatter appears minimal in the plot with C;(k) for the interval of k between 0 and 1/2. For the interval of k between 1/2 and 1, the scatter is much less on the plot between C;(1-k) and k. Therefore, it is suggested that when determining a value of C; for a given value of k, the plot of CL(k) versus k be used for values of k less than 1/2 (except for k close COMO) pani sehiey plotmor ey Glog). versus) kebe) used stor values vot ke vereiat en, cham 1/2 (Gxeeave stor k ClOSe wo il))) (SES Figs 4s) or l< close tO O45 ae can be assumed that C,; = 4.5. However, for k ~ 1, the value of Cy, can vary from about 4.5 to zero, since as the clearance is increased from the point where $ = 90° and k = 1, both @ and k remain at their maximum values of 90° and 1, respectively, while the lift effect diminishes to zero. When the above relationships between >, k, C,, Cj, (1-k), and Cy (k) are plotted for only the 4-inch-diameter pipe model, the scatter is reduced. Although the data for all three diameters completely overlap (showing the same relationships hold for all diameters), the amount of scatter increases with the smaller diameter models. This is because the data extend to higher relative clearances (clearance-diameter) for the smaller diameter models than the corresponding data for the 4-inch-diameter model, since all models were tested at the same actual clearances. Since the lift effect diminishes at high values of the relative clearance, the lift forces on the smaller diameter models at the largest bottom clearances were very small in many cases. This is especially true for the smaller waves and higher orientation angles, where the horizontal velocities perpendicular to the pipeline were very low. In such cases, they litt forces were often inswenifieant ; so;the values of Cy> o; and k calculated from the least squares analysis were not as accurate. In addition, as the lift forces decrease with high relative clearances, eddy-induced forces may approach the magnitude of the lift forces, thus introducing further error in the calculated values of C,;, $, and k. The lift forces were generally significant for all clearances tested using the 4-inch-diameter pipe section, and since the measured forces were larger, the experimental error involved in measuring them was less than for the smaller diameter models. Because of this, the data taken for very large bottom clearances were not included in the plotted relationships. For higher clearances, values of k and $ equal to 1 and 90°, respectively, would be expected, since the choking phenomenon would not occur throughout the vave cycle. However, as the clearance is increased, the lift effect diminishes, resulting in decreasing values of the coefficient of lift. 87 ‘y snszea (4) 19 20 (4-1) 5 ‘Op 9rnsTy CA-TI@19 x@19 YO 88 an x VIO wIi-b X vig wi-e V UJLIWVIO NWI-2 © Ss C(y¥-1)307) <— ——> ye) If such data were included in the plots of C, versus k and ¢$, values of C; ranging from 0 to the maximum values shown in Figures 40 and 41 would be present in the vicinity of k = 1 and ¢ = 90° in the respective plots. The same applies to the plots of C, (kK) versus k and 9. These trends were observed in the data taken for the largest bottom clearances (1 and 2 inches). However, since these lift forces were so small, a significant amount of error could be introduced into the cal- culated values of C;, $, and k because of the presence of eddy-induced forces, as discussed above. Therefore, these data were omitted from the plotted relationships, since errors in $ or k corresponding to low values of C; would produce considerable scatter, obscuring the valid relationships shown. 4. Relationships Between ¢ and k and Parameters Defining the Wave and Pipeline Conditions. To use the above relationships between Ci, od, and k to determine the wave-induced lift forces acting on a pipeline, either $ or k must be known. Thus, a value of one of these parameters must be determined from relationships of ¢ or k with the wave conditions and pipeline con- figuration. The lift force phenomenon is a function of the following variables: (a) Pipeline configuration (1) Diameter (2) Clearance (3) Orientation angle (b) Fluid properties (1) Density (2) Viscosity (c) Wave-induced flow conditions (1) Maximum horizontal water particle velocity perpendicular to the pipeline axis (2) Wave period, which represents the duration of the flow in one direction (3) Length of the horizontal excursions of the water particles perpendicular to the pipeline axis (this quantity is di- rectly proportional to the product of the above two param- eters) Assuming that only water with a limited range of temperature is being | dealt with, the fluid properties will be ignored for the present. The orientation angle of the pipeline can be handled as discussed above, considering only the components of the horizontal fluid motions 89 perpendicular to the pipeline axis. Since the length of the horizontal water particle excursions is directly proportional to the product of the wave period and the maximum horizontal water particle velocity, only four independent variables are left: diameter, clearance, horizontal water particle velocity, and wave period. Thus, any single parameter used to relate Cy» p, or k to the wave and pipeline conditions must include these four variables. This constraint is necessary if the rela- tionship is expected to be valid for general application under any set of wave and pipeline conditions. The four variables can be arranged into several dimensionless param- eters, The important parameters should include the following: (1) relative clearance, clear/Dia where clear = bottom clearance Dia = pipe diameter (2) Keulegan-Carpenter parameter, Oe T/Dia where T = wave period U ax 7 Component of maximum horizontal water particle velocity perpendicular to the pipeline axis (3) clear/u T NOTE.--Not all of these parameters are necessary to describe the system since some are redundant, but some may be more useful than others. Since viscosity is an important variable involved in the choking phenomenon, the Reynolds number, u,,, Dia/v, and a Reynolds number for the clearance, u,,, clear/v, are also important parameters (where v = kinematic viscosity). The dimensionless parameters, clear/upa, T, Wage T/Dia, hive clear/v, and nex Dia/v, were plotted versus the lift force parameters, Cy o, k, Cy (1-k), and C, (k), for constant values of the relative clearance, clear/Dia. The correlation was not good with the parameters involving the coefficient of lift (C,, C,(1-k), and Cy(k)). However, good cor- relation was found between several of the dimensionless parameters and the quantities > and k. The parameter, clear/u,.x T, exhibited the best correlation with both » and k for each relative clearance, although there was some varia- tion in these relationships for the data corresponding to the different pipe diameters (see Figs. 47 to 52). Although the differences are not large, the data do indicate the presence of a scale effect in these relationships. 90 xPePul ‘TojJoWeTp YOUT-p OF bo snsioA > “Lp oansTty CLOXVWNIsSYVI1I . oo°¢ z poe s Z 2-03 > YZ, ¢-0! 00°9 d x x By Xe is x x< X x x 2 025 o m 07° est O 09 9T/t (a +9/€ © 237% (Gl) V oe x vV1I0/8v319 Ge ‘o) 48 ecest Qo cote § 9| xXeUul L n IeoyTo “L9JOWETP YOUT-¢ OF | SNSIOA $ ‘“Qp oIn3Ty (LOXVWNI/SYV3I19 oo°*¢e z y—-O! Ss Z 2-03 Ss 2 ¢-Ot 00°*9 atst ozs ¥ x pe ae 4 Ys Cost Qo 9 ge YY OSZz/t vid/av3i3 xd0¢009 00s SIZ xeu it n Ieopo *ZoVOWeTp YyOUT-z LOFT ] snsioa > ‘6p 9aIn3TY CLOxVHWNI/Yv379 Teor S z se! S z ceo, <0l0h9 o/t ° o/t o) ZE/E erst (9) zest LOtst vig/suv3i19 xXxd0G090 oot 93 xeUw L n Ie9TO *LoJOUeIp YOUT-p IOF ] SNSIOA Y ‘0S oIN3TY CLOXVWNI/8VI12 oo°s 2 1-0} S z 2-08 S 2 a Sacre Sm ae eer (ce aera) (on Poem) fy eee a fromm Coa) foe ad Fae em OR est S 9 $9 oe zest Oo (+) 69/t Oo xXxW0Oe09 eecst V1I0/8V319 94 yal “IoYOWeTp YOUT-¢ OF L Xeul iat a SNSIOA Y “TS oINn3Ty CLEXVWNI/YV3I1)9 2-9! Ss 2 9/3 zyv/t 93/3 bz/t Obst as2z/t vVI0/8v313 xId0Oc090 JS) Tent xeUul L n Ie9TO *LoVoUe Tp YyOUT-Z OF | snsIoA Y "7S oaInsTy CLOXVWNI/AVI19 S z 7-01 S o/% e/t ZEe/e gts ze/t 4gatst vid/uvai9 xd0e09 96 » and k were also correlated with the Keulegan-Carpenter parameter, Unax I/Dia. However, these relationships were not the same when the data corresponding to a given relative clearance were compared for dif- ferent pipe diameters. The relationships were the same for a given absolute clearance, rather than a relative clearance (clear/Dia). These relationships are shown in Figures 53 and 54 for the combined data from all three pipe diameters. The parameter, Ula clear/v, demonstrated correlation with both 6 and k, but these relationships also exhibited a scale effect, such that the relationships for a given relative clearance were not the same when comparing the data for different pipe diameters. Figures 55 and 56 are examples of these relationships for the 4-inch-diameter pipeline. Correlation between the Reynolds number, ung, Dia/v, and the param- eters, » and k, was not good, especially when comparing the data for the different pipe diameters. Since none of the above dimensionless parameters alone could be used to determine a value of > or k for any given pipe diameter, clear- ance, and wave condition due to the presence of scale effects, several. of the parameters were combined in various ways to form different dimen- sionless parameters containing all four of the important variables (clear, Dia, unax, and T). An attempt was made to find a single param- eter containing all of the important variables that was well correlated with » or k for all wave conditions, pipeline sizes, and configurations. Several relationships were found that exhibited good correlation for all the wave and pipeline conditions tested. However, since this is a model study and, therefore, limited to lower values of the Keulegan- Carpenter parameter and Reynolds number than prototype design situations in the ocean, caution should be used in extrapolating these results. The dimensionless combination, (clear/u,,, T) ia/ujg,T), demon- strated the best correlation with both » and k for all conditions tested. These relationships are given in Figures 57 and 58. Since both k and » define the point at which choking occurs in the wave cycle, it appears that the choking phenomenon is directly dependent on the water particle excursions relative to both the pipe diameter, (Dia/u,., TE) 5 and the bottom clearance, (clear/u,,, T). Although the parameter, (clear/ aye T), is equivalent to the ratio of the bottom clearance to the horizontal excursion of the water parti- cles (differing only by the constant 1/7), the quantity (u,,, T) should not be thought of only as defining the length of the water particle excursions. Both variables, u,,, and T, are independently important in defining the choking phenomenon. The larger “max? the sooner the chok- ing conditions will develop in the wave cycle for a given clearance and pipe diameter. Similarly, since the wave period, T, defines the duration of the horizontal flow in one direction, the larger the wave period, the sooner choking will develop relative to the temporal length of the wave cycle. SMe ot WI ost WI 673 MI 93/76 WI @73 WI O373 14 300°0 ZINVYEVIID xqI0¢600 | etd ae SNSIOA Y *Sg omnsTy VIGO/CLOXVWN) 98 xXeul | etd Ak n | snsion > “PG sInsTy VIGQ/CLOXVWAD oT 6 8 L 9 s % € z t 0 NE 2/f WI 67% WI 9t/€ NI @/3 Wr 9t73 LJ 100°9 JINVIVIII xXxd0¢090 99 A xeul ‘LoJoWeTp YOUT-p TOF Ieoypo n ] SNSIOA Y °Sg Ansty ISTA/CYVIIIEXVWN) 0o0°e Z co" 8 9 » z 201 @ 9 » ? (o) ove Oo Vv ert Bo Wy © O et/t Ve 3 % zest sort ccest QO RAO via/suv312 xd0¢09 100 “IOVOUWETP YOUT-p TOF g SNSIOA d “9g OINSTY xeul Ieapo n ISIA/CYVIIISKVWN I) o00°¢€ 2 eo" 8 :) : 2 20T @ 9 > ? yo! o est QO 03 Oo Ce 4 9T/t O 9 oO d r+) ¢9/€ © Og r &, zevt wy 69/1 VY 08 7) ®& ECE7R sk . © QW vVIO/¥v319 oe) 19) 00T eee Ee etd Iesqto CLOXVWA/VIGIECLEXVHN/YVIID) 102 vid WI-b X Voxeax x & vid wWi-e V xx YILINVIG WI-2 © xeul xeuL is | 4 snsioA > ‘gS OINSTY Iesypo CL@XVWN/VIOI@CLOXVWA/YV3II) 2 y-03 S z 2-01 S 2 ¢-01 S z ,-08 Id €930) yx ee om vig «wi-s x vid Wie walauvid wiz x 00t o.aXx 103 The slight amount of scatter in these plots in the vicinity of k = 1 and » = 90° is due to the error in calculated values of ¢ ‘and k for the largest bottom clearances where the lift effect was small (as dis- cussed above). Larger values of:the dimensionless combination, (clear/u,,, T) (Dia/u,,, 1), than given in the plots would correspond to larger bottom clearances and pipe diameters relative to the maximum velocities, wave periods, and water particle excursions. For these conditions, the values of k and ¢ would remain at 1 and 90°, respectively, while the lift effect would eventually diminish to zero with increasing values of this parameter. These trends are evident in the data taken at the largest bottom clearances (1 and 2 inches), although these data were not included in the above plots. Similarly, lower values of the dimensionless parameter than given in the plots would correspond to higher maximum velocities, wave peri- ods, and water particle excursions relative to the smallest bottom clearances and pipe diameters. So for lower values of this parameter, both k and » should remain at their defined minimum values of 0 and 0°, respectively, corresponding to lift forces acting in the upward direc- tion only, with very little or no flow possible under the pipe section. Although » was defined as varying from 0° to 90° only, negative values of @ are exhibited in the data for the lowest values of the dimensionless parameters plotted. However, since most of these data points correspond to the smallest diameter pipeline model tested (2 inches), this could be partly due to experimental error, since the measured forces were smallest for the smallest model. Also, part of this dis- crepancy could be due to the difficulty of accurately defining the peak of the wave crest in the experimental wave records. This point was arbitrarily defined as the midpoint of the zero crossings on either side of the wave crest in the digitized data records. However, in some cases, the waves were not perfectly symmetrical, so the maximum eleva- tion of the water surface did not coincide exactly with the midpoint of the zero crossings. This was especially true of the largest waves with the longest periods, which in the plotted relationships would correspond to the minimum values of the dimensionless parameters (at the lowest bottom clearance tested). Thus, the actual kinematics under these waves would be slightly out of phase with the calculated kinematics, resulting in an error in the calculated value of ¢. However, this source of error should be the same for the large-diameter models as for the smallest models. 5. Relationships Between > (clear/Dia) and k (clear/Dia) and Parameters Defining the Wave and Pipeline Conditions. Many other useful relationships were found by multiplying > and k by the relative clearance, (clear/Dia), and plotting these dimensionless products versus various dimensionless parameters defining the wave and pipe conditions. Figures 59 to 62 are examples, although several other parameters also showed good correlation with » (clear/Dia) and k (clear/ Dia). 104 2-95 Ss xeul | cal SNSIOA al °6S omnsTy | resto Ie9TO Iv9TO CVIO/YVIIIIGCLEXVWN/YVI1II Zz eg OSS Zo Ue Ss z e201 vid vid ¥ILIWVIG WI-% WI-€ WI-2 s x Vv o CVIG/YVAIIIIEGN 105 | xeul etd di n etd : a | Sues lees? Os) uur CVIO/YVIIIICCLEXVWN/YV3I193 ta Ot eS ze gio 5 2 g-0% vid vida yaLIWwvid WI—b WI-€ WI-Z S x Vv © CVIG/YVIIIIGING (930) 106 2-01 S ana ii xeu i xeur ana ea baa =Seide/-e oe oe pees Ls CVIO/YVIIII@CLOXVWA/YV3IIIISCLEXVWN/VIOILYDS oe Oe tes 22. Mee ONe mS 2 efi vid vid ¥Y3LIWVIO “NI => NI=-€ NI=-Z x Vv + z20° 90° Pa ° CV¥IO0/9YWI1IIeN @ = zz° 92° o€e* 107 Ss .|_ PTd Ieoeyto xXeUul It, n Iesyo SNSIOA etd Ieayto ie 79 eIns Ty CVIO/YVIIIIOCLOXVWN/YVIIIIPCLOXVNN/VIOILNOS on S- 2 ¢ 01 Ss yell Ss eA vida vid yZLIWVIO WI-b WI-€ WI-2 x Vv © CVIO/YVIIII@ING zt, 930 + ¢ 93 108 Both » (clear/Dia) and k (clear/Dia) are correlated with the dimen- sionless combinations (clear/u,ax T) (clear/Dia) and vDia/uyax ily (clear/u, x T) (clear/Dia). However, k (clear/Dia) appears to be better correlated with the first parameter, while » (clear/Dia) shows better correlation with the second parameter. It is clear that for values of the dimensionless parameters lower than those shown on the plots, both > (clear/Dia) and k (clear/Dia) will remain at a value of zero. This would correspond to situations where the clearance was minimal relative to the horizontal velocities, wave periods, and horizontal excursions of the water particles. Thus, both k and ¢ would be expected to equal zero and 0°, respectively, and the relative clearance would either equal or approach zero. Large values of the dimensionless parameters correspond to situa- tions where the clearance is large relative to the horizontal veloci- ties, wave periods, and horizontal excursions of the water particles. For these cases, k and ¢ will remain at maximum values of 1 and 90°, respectively, while the relative clearance, (clear/Dia), will increase with increasing values of the dimensionless parameters. But as the relative clearance is increased beyond this point, the lift forces will decrease to zero, so extension of the plotted relationships to much larger values of the dimensionless parameters is of little value. 6. Relationships Between the Coefficients of Lift and Parameters Defining the Wave and Pipeline Conditions. The coefficient of lift, Cj, the effective positive coefficient of Us, Clas) 5 iS Gietoctiwe MEpEES Comesaueioms ore Ibu, Cilio) 5 eiocl the maximum effective coefficient of lift (maximum of C, (1-k) or Cy (k)) were plotted against various combinations of the dimensionless param- eters. The parameter, (clear/u,,, T)(Dia/umgxT), which previously gave the best correlations with » and k also demonstrated the best cor- relation with Ch. Cy (1-k), and Cj,(k). However, these relationships exhibited more scatter than the previously discussed interrelationships between the coefficients of lift and the parameters, k and $, so it is suggested that the previously discussed relationships be used for design purposes. 7. Relationships Between the Lift Forces and Parameters Defining the Wave and Pipeline Conditions. As with the coefficient of lift, the total lift force (ey = 1/2 C; p A pine) can be partitioned into the maximum positive lift, FLU} and the maximum negative lift, F,(k) (Fig. 6). These three forces, as well as the maximum lift force (maximum of either F, (1-k) or F, (k)) were plotted against various combinations of the dimensionless parameters. Only one relationship exhibited good correlations for the data from all three diameters plotted together. This was the Reynolds number, Upa,Dia/v, versus the maximum lift force (either FrG@ch) sor F,(k), whichever is greater) (Fig. 63). 109 00002 “toqunu sproudey 9yi snstoa (dAT}eSou IO oATAtSOd) edzZ0F AZTT unutxey ISITA/CVIC@XVWN) oo00et oo00gT O00vT oo002t oooot 0008 0009 O00» "€9 aansTy \v7 = \v/ VIO wNI-¢ 7 VIO wNI-€ Vv YILINVIO NE-2 xX Vv @ CH-TI@713 YO N14 JO WAWIXVW Ke) This relationship shows that for any pipe diameter, orientation angle, or bottom clearance, the maximum lift force increases with the Reynolds number in a regular manner, at least over the range of the data in this investigation. The maximum lift force may occur in either the upward or downward direction, depending on the magnitude of the bottom clearance relative to the wave conditions and pipe size. This relationship does not hold for the maximum upward lift or maximum downward lift alone, but only for the largest of these two forces in any given situation. 8. Relationships Involving the Vertical Coefficients of Mass and Drag and the Vertical Inertial and Drag Forces. Both the vertical coefficient of mass and the vertical inertial forces were plotted against several dimensionless parameters defining the wave and pipeline conditions, but no useful relationships were found. This is not surprising when considering that the vertical iner- tial forces are relatively small, and thus subject to error from the transverse eddy-induced forces which were not accounted for in the least squares analysis. No attempt was made to plot relationships involving the vertical drag forces or drag coefficients, since these forces were negligible. 9. Relationships Between the Horizontal Coefficient of Mass and Parameters Describing the Wave and Pipeline Conditions. A limited number of horizontal force data were taken using the 4-inch- diameter two-dimensional model. Values of Cy and Cp were calculated from the least squares analysis, and an attempt was made to relate these coefficients to various dimensionless parameters describing the wave and pipeline conditions. Figure 64 shows the horizontal coefficient of mass plotted versus the relative clearance, clear/Dia, together with the potential flow solution for a circular cylinder in the vicinity of a plane wall subject to a uniform flow with constant acceleration (Grace, 1974). The data follow the potential flow solution reasonably well, although for a given rela- tive clearance, there appears to be some variation in the value of Cy with varying wave conditions. Also, the wave force data give slightly higher values of the coefficient of mass for the highest bottom clearances tested. Although the experimental data are limited, they indicate that the potential flow solution may be very useful in determining a value for the horizontal coefficient of mass, at least for wave conditions where the inertial forces predominate over the drag forces. However, Since there was some variation in the values of Cy for different wave conditions for the same relative clearance, an attempt was made to determine relationships between the horizontal coefficient of mass and the various dimensionless parameters defining the wave and pipeline conditions. Reasonably good correlations were found between “uoTJeIOTOONe YUEPSUOD YIM MOTZ e@ IOF ATOOYI MOTF TeTRUSJOd YITM Wy ,equoztzoy oyi FO uostaedwojn “pg omnsTy viags¥vail9) s° 6° €° r A Te 0 bale y 3 V 6°2 ‘Oy AMOZHL MOTS IWILNILOd vy aA 7 a a ie = v\ 78°23 Vdc “AN Vern V ze g* several of the parameters. Figure 65 shows the relationship for Cy versus GIS Ebe/ Vasc! c 10. Relationships Involving the Horizontal Coefficient of Drag. The horizontal coefficient of drag was plotted against several dimensionless parameters, but no useful relationships were found. This was expected since the horizontal drag forces in this investigation were much smaller than the inertial forces, due to the limited horizontal excursions of the water particles relative to the diameter of the pipe- line. idl Example Problems. GIVEN: A design wave with height, H = 10 feet and period, T = 10 seconds acts on a pipeline with a diameter, Dia = 8 feet in a water depth, d = 80 feet. The pipeline is oriented at an angle of 30° with respect to the wave crests. Section A of the pipeline is in contact with the bottom; section B spans the bottom at a clearance, clear = 6 inches. FIND: For both sections A and B, find (a) the values of the lift force parameters (C;, >, and k); (b) the maximum positive and negative lift forces; (c) the positions of these maximum lift forces in the wave cycle; and @iithe att! force at Gy=) 0s anithe wave! cycle. SOLUTION: in = S222 Be CO)? SSI Ree OO eel ; @d =. 80 _ es = SD = 0.1562 Using tables es 0.1885, so L = eee 424 feet 2 Ik j ‘i 0.1885 sinh ote 1.481 z = distance from bottom to center of pipe sections. 3 | ah xXeul n | SNSIOA Ny [equoztzloy °S9 ean3sTy reayTo CLEXVWNI/YV3I19 2 2-91 Va) V V Yo Ww V v Ve Va Tey, a ¥ a Vy V 7 WU V g-o03 W4 For section A (clear = 0) ZV AS Cet == — 0.00943 From tables, cosh ote LOO 7 ee (10) (1.0017) Unaxs pte Se Sp eee “(i0y (481) 2.12 feet per second sinh (>—) Component of Umax perpendicular to the pipeline axis is Unax (cos 30°) = (2.12) (0.866) = 1.84 feet per second (a) Since the pipe is in contact with the bottom, (clear = 0), ¢ = 0° and k = 0. From Figure 40, C;, = 4.5. (b) Maximum positive lift (per unit length) F, (1-k) i ! | 1 2 7) Cy, Oo A Unax (1-k) (iS) (2) (2) Gls BQ)? G0) Nie = 121.9 pounds per foot. Maximum negative lift (per unit length) Sinmeenies—) Oh mehere msi mommneigatavemlistt, sandatielestts force; as positive throughout the wave cycle. (c) Since » = 0°, the positive lift forces are maximum at 0° and 180° in the wave cycle (under the crests and troughs), corresponding to the points of maximum horizontal velocities. The lift does not become negative, but diminishes to zero at 90° and 270°, the positions of horizontal flow reversal in the wave GVEIS. (@) Ne 8S 120 Ete) =p) 0) A Uns Wicossu(Ghe=n ge =" ic] (4.5) 2) Ga8= eos” (120° = O°) =O) = 30.5 pounds per foot Ee) For section B (clear = 6 inches) z= 4.5 feet z= +> = 0.0106 424 From tables, cosh at 1.0022 _TH cosh (7 Eeu@lo)Gco022) ay eee : Chiebe y TGP sine “Choy aany = feet per secon component of u,,, perpendicular to the pipeline axis is Up. (COS S07) = (2.15) Ws3s0) = 1.84 cece por second (a) Use Figure 57 to determine a value for k clear Dia (0.5) (8) TT (| Cee UL ayy Go) 0.0118 SO ieeom Papuee 57, k = O67 fo} and from Figure 58, $ = 45 Alternatively, either ¢ or k could be determined from Fig. 39, once the other is known. From Figure 46, for k = 0.67, C, (1-k) = 25795 : 2 Sen so Ch = sae = bk (b) Maximum positive lift (per unit length) Il 2 F, (ik) Cry OwAVU Ls ni(lels) (8.3) 2) @) G1. 84)" G@ = 0.67) = 74.2 pounds per foot 116 Maximum negative lift (per unit length) 1 - Fy (k) = - 5 Cy P A Umax” (K) 1 5 EQ) QQ = (O.67) - 150.6 pounds per foot (c) Since 6 = 45°, the positive lift forces are maximum at 0° + 45° = 45° and 180° + 45° = 225° in the wave cycle, and the negative lift forces are maximum at 90° + 45° = 135° and 270° + 45° = 315° in the wave cycle. (cl) Ae @ = 120° Fea GC, 6 Atha. (eos> C205 450) = 0667] NJR Ne (8.3) (2) (8) (1.84)? Iicas™ GI2Z02 = 25°) = 0.67) = - 135.6 pounds per foot Again, it should be stressed that the relationships involving the lift force parameters, Cy, ¢, and k, were determined from model studies conducted at much lower values of the Keulegan-Carpenter parameter and Reynolds number than those encountered in full-scale situations in the ocean. Therefore, caution should be used in extrapolating these results to prototype designs. Further studies using a larger scale facility are necessary to evaluate the importance of scale effects in these relationships, to determine their limitations, and possibly to extend or modify them so they are valid for any scale. TV. CONCLUSIONS 1. The traditional steady-flow lift force model, expressed as a = W/Z Gi, OIA u*, is not a suitable model for the description of wave- induced lift forces. This model assumes that the lift force acts in one direction only (upward or downward) throughout the entire wave cycle. 2. For pipelines located at a small clearance above the bottom, a viscous choking effect limits the maximum velocities through the constriction formed by the bottom clearance. Correspondingly, the pressure drop on the bottom side of the pipe section is also limited. In contrast, the flow velocities and corresponding pressure drop over the top side of the pipeline are not limited. As the choking effect develops and the flow becomes restricted through the bottom 117 clearance constriction, more of the flow must be diverted over the top of the pipe section, resulting in a downward shift in the stagnation point, as well as an increase in the flow velocities and associated pressure drop over the top side of the pipeline. The induced changes in the flow pattern, velocities, and associated pressure distribution over the pipe section due to choking through the bottom clearance constriction result in an upward lift force, rather than the downward lift force predicted by potential flow theory. 3. Thus, for an oscillatory wave-induced flow, the lift force acts downward in those parts of the wave cycle where the horizontal water particle velocities are not high enough to produce choking through the bottom clearance, In this case, the unrestricted flow is faster through the bottom clearance constriction than over the top of the pipe section, so the corresponding pressure distribution results in a negative lift toward the bottom boundary. However, in those parts of the wave cycle where the horizontal velocities are sufficient to induce choking through the bottom clearance constriction, the lift force acts in an upward direction. 4. For a given pipe diameter and wave condition, as the bottom clearance is increased, higher velocities are necessary to produce the choking effect. Thus, the negative lift force can reach a greater magnitude and occur later into the wave cycle before the choking condi- tion is induced. Correspondingly, the positive lift that occurs only after the choking condition develops is limited to a smaller part of the wave cycle, and the maximum magnitude of these forces decreases with increasing clearance. In addition, since there is a small timelag involved in the development of the choking phenomenon and the transition from negative to positive lift, the maximum positive lift occurs later into the wave cycle, although its magnitude is diminishing. 5. All major features of the wave-induced lift force phenomenon can be described adequately by a modified lift force equation, Fy = Wie Gy, OA We [cos* (@ - ) - k], where @ represents a phase shift in the position of the maximum positive (upward) lift force relative to the point of maximum horizontal velocity at the center of the wave crest, and k represents the proportion of the total lift force cycle that acts in the negative (downward) direction. The values of $ and k vary from 0° and 0, respectively, for the case of a pipeline touching the bottom, and increase with increasing clearance (for a given pipe- line and wave condition) to maximum values of 90° and 1, respectively, when the pipeline is far enough from the bottom so that the choking condi- tion does not develop. oy = 0.) andiky =) 0) \comrespond jtol lissemtoncesm that are positive throughout the wave cycle, with maximums occurring at the points of maximum horizontal velocity under the wave crests and troughs. @ = 90° and k = 1 correspond to negative lift forces throughout the wave 118 cycle, with maximum downward forces occurring under the crests and troughs of the passing waves. These two cases represent the extreme conditions bounding the lift force phenomena. At any intermediate clearance between these limiting cases, both positive and negative lift forces will occur at different parts of the wave cycle, and the positions of the maximum upward and downward lift forces will not coincide with the positions of maximum horizontal velocities in the wave cycle. In order to use this lift force model, values of the parameters, Cy, >, and k, must be determined for the given set of wave and pipe- line conditions. A model investigation was carried out to determine relationships between these parameters and various dimensionless param- eters defining the wave and pipeline conditions. 6. A direct relationship was found between the lift force parameters, > and k. Relationships were also found between the coefficient of lift, Cy; and both ¢ and k. In addition, C; can be partitioned into the positive effective coefficient of lift, Cr (1-k), and the effective negative coefficient of lift, C,(k). Both of these parameters are also related to both > and k. The correlation is better with k than ¢ for the relationships involving C;, Cy,(1-k), and Cj, (k). All of these relationships were the same for all pipe diameters, bottom clearances, and wave conditions tested. 7. The average value of C; at k = 0 and 9 = 0° (which corresponds to a pipeline in contact with the bottom with no clearance) is 4.5. This is the same as the potential flow solution for the lift force on a circular cylinder against a plane wall subject to a steady, inviscid flow parallel to the wall. 8. Maximum values of Cy; occur at k = 1/2 and $ = 30°, where Gu-eoee Invche imterval strom ky— 0) toy 1/2 and id) — Oy to 30, athe effective positive coefficient of lift C;(1-k) remains at approximately 4.5, while the effective negative coefficient of lift C, (k) increases ROMO McOn4 San inethie amtenval: ron ki 1/2 to land m=50c) topo. Cy (1-k) decreases to 0, while Cy;(k) increases to reach a maximum of about 6 or 7 at k = 0.75 and = 45°, and then decreases to a maximum of 4.5 ate ike i ame @ Ss 90°. 9. Using the above relationships between Cy, », and k, if either g or k is known, the remaining two parameters can be determined. Therefore, an attempt was made to find relationships between 9 and k and various dimensionless parameters defining the wave and pipeline conditions. The best correlation was found in the relationships between @ and k and the parameter clear/u,,,T for constant values of the relative clearance, clear/Dia. However, comparison of the data corresponding to the different pipe diameters indicates a slight scale effect is present. 119 @ and k were also related to the parameter u,,,clear/v for constant values of clear/Dia, although the scale effect was worse for these rela- tionships. ¢ and k showed very good correlation with the Keulegan- Carpenter parameter, u a T/Dia, although these relationships were the same for a constant absolute clearance, rather than a constant relative clearance. Correlation between > and k and the Reynolds number was poor. 10. Because a scale effect was evident in the above relationships, several of the dimensionless parameters were combined to form new dimensionless parameters that contained all of the important variables (clear, Dia, ugayx, and T). An attempt was made to find a single param- eter that was related to either » or k for all wave and pipeline condi- tions tested in this investigation. Both ¢ and k showed very good correlation with the parameter (clear/u,, T) (Dia/u,,,,1). These relationships were valid for all pipe diameters. bottom clearances, orientation angles, and wave conditions tested. In addition, the relative clearance was combined with both ¢ and k to form the quantities $(clear/Dia) and k(clear/Dia), both of which exhibited very good correlation with more of the dimensionless combina- tions than either or k alone. k(clear/Dia) was best correlated with (clear/u,,,T) (clear/Dia). o(clear/Dia) was also correlated with this parameter, but exhibited better correlation with the parameter vDia/UmaxT (clear/u,,T) (clear/Dia). Like Gyo Cyd), ead Cy (k) were correlated with the same parameter as o and k, (clear/u,,,T) (Dia/u,,,,T) - However, these correlations were not as good as the previous correlations between the coefficients of lift and k or $. 12. For a pipeline that is not parallel to the wave crests, the lift forces are apparently due only to the components of the horizontal water particle velocities perpendicular to the axis of the pipeline. Using this convention, consistent values of the coefficient of lift, Cy, are obtained for all angles of orientation. In addition, the rela- tionships between the lift force parameters Cy, », and k, as well as relationships between these parameters and various dimensionless param- eters defining the wave and pipeline conditions, are identical for all angles of orientation. 13. The maximum lift force (F;(1-k) or Fy, (k), whichever is greater) exhibited good correlation with the Reynolds number, (UnaxDia/v). This relationship did not hold for the maximum positive lift (Fy, (1-k)) or the maximum negative lift (F,(k)) alone, but only for the largest of these two forces in any situation. The relationship was the same for all diameters over the range of conditions tested. 120 14. The horizontal coefficient of mass, Cy, showed excellent agreement with the potential flow solution for a circular cylinder in the vicinity of a plane wall subject to a uniform flow with constant acceleration. These results indicate that the potential flow solution may be useful for selecting a value of Cy for wave-induced forces, at least for situations in which the inertial forces predominate over the drag forces. The horizontal Cy was also correlated with several of the dimensionless parameters defining the wave and pipeline conditions, such as the parameter clear/u,,,T. V. RECOMMENDATIONS FOR FURTHER RESEARCH 1. Experiments similar to this investigation should be carried out in a larger wave tank facility. This would allow the testing of larger diameter pipeline models as well as experiments at higher Reynolds numbers and higher values of the Keulegan-Carpenter parameter. Such an investigation is necessary to determine the validity of extrapolat- ing the results of the present study to design situations in the ocean, and to point out any weaknesses or limitations of the proposed lift force model due to scale effects. 2. It would be of interest to perform experiments to evaluate the magnitude, phase, and frequency spectra of the vertical transverse lift forces due to eddy shedding for a horizontal cylinder subject to oscillatory horizontal flow velocities. This could be done by oscil- lating a test cylinder horizontally in still water away from a boundary, or by using a pulsating flume facility. The horizontal flow patterns at the bottom could be simulated, but without the lift force phenomenon due to the boundary. Only the transverse lift forces due to eddy shedding would act in the vertical direction, so the magnitude and time history of these forces could be easily measured. A thorough analysis of the eddy forces for different pipe diameters and flow conditions would allow an evaluation of their importance rela- tive to the Bernoulli-type lift forces, and at the same time explain some of the variations in the vertical wave force parameters calculated from an analysis which neglected the eddy forces because they could not be separated analytically because of their random nature. Adequate knowledge of the eddy forces would allow the addition of the eddy lift force term, Fj, = 1/2 C{ pA Ula to the Morison equation with appro- priate values of the coefficient C{ for any given set of wave and pipe- line conditions. It should be noted that evaluation of the eddy forces for a cylinder away from a boundary would only give an approximate estimate of the eddy release phenomenon for a pipe located near the bottom. The presence of the bottom boundary changes the flow pattern, velocities, and pressure distribution around the cylinder, and therefore would be expected to have some effect on the formation and release of eddies. 12| 3. Since the restricted flow through the narrow bottom clearance constriction is the critical part of the lift force phenomenon, the effect of pipeline roughness and bottom roughness on the wave-induced lift forces should be studied. This has practical significance, since the ocean floor is not necessarily smooth, and pipelines installed in Marine waters may soon become encrusted with marine organisms, thus increasing their surface roughness. 4. The effect on the lift force phenomenon of a horizontal bottom current superimposed on the oscillatory motions of the wave action should be investigated. 5. The effect of porosity of the bottom on lift forces should also be investigated. 122 LITERATURE CITED AL-KAZILY, M.F., "Forces on Submerged Pipelines Induced by Water Waves," HEL 9-21, Hydraulic Engineering Laboratory, University of California, Berkeley, Calif., Oct. 1972. GRACE, R.A., ''Wave Forces on Submerged Objects,'' Report No. 10, Look Laboratory, University of Hawaii, Honolulu, Hawaii, July 1974. MORUSONR RIM eteal.. | ihliethorcesp exerted by Surtace Wave's) on Palie's), Petroleum Transactions, Vol. 189, 1950, pp. 149-154. PAULLING, J.R., and SIBUL, O.J., "Instrumentation for Digital Recording," Proceedings of the 15th Amertcan Towing Tank Conference, Ottawa, Canada, 1968. YAMAMOTO, T., NAYH, J.H., and SLOTTA, L.S., "Yet Another Report on Cylinder Drag or Wave Forces on Horizontal Submerged -Cylinders,"' Bulletin No. 47, Engineering Experiment Station, Oregon State UmatvierSulttyiComvyalliash Onege eApia. lS. 123 Pg APPENDIX A LEAST SQUARES ANALYSIS OF EXPERIMENTAL DATA Using Morison's method for the calculation of wave forces on a pipeline, the vertical component of the wave-induced force may be expressed as: a9) < | = ! Cire) Rena Ch) aan btar ie ov Cy p V ae o LZ Gp OWN VW |v| I + W/2 Gi @ I Uae [case (@ = ®) = k] 1 120. OM une eee ey Since the transverse lift force associated with eddy shedding (FL) is a random phenomenon, there is no way to handle its time history in analyzing a wave force record with several other forces occurring Simultaneously. Because the Bernoulli-type lift forces were much larger than the eddy-associated forces for a pipeline located close to the bottom, the eddy-associated lift force term was dropped from the analysis. The vertical components of the water particle velocities and accelerations near the bottom are small in comparison with the corre- sponding horizontal components. As a result, the vertical lift forces due to the horizontal components of the water particle velocities are generally much larger than the vertical drag and inertial forces. The drag forces are especially insignificant since the vertical excursions of the water particles near the bottom are smaller than the diameter of the pipeline. Using linear wave theory, the kinematics of the wave-induced water particle motions with respect to time can be expressed as: u= TT a POTS cos 0 (A2) sinh (—— L D TH Samia Cr WSS “aE TL sin 0 (A3) sinh are 125 where dv rei. an ot 1 nae es wave height wave period wavelength stillwater depth vertical distance above the bottom 271t T = —— = position of the wave cycle with respect to time. (A4) Substituting these expressions into the vertical force equation yields: ey, Cosel = Gp, Sted |sime| > Chip feos (@ = 0) =i. : 271Z Eianipon alsoW2 nati sami (Fr) Min, 2 : 271d W sinh e) Nee CoM TIZ: Ee oAt*H? Suan” (or) = Gp pall Soe RES Me gagihe (4 2 271Z » Gr oAT*H? cosh” (aT) 2T2 ~— sinh? 5 or Py i M Mv where i 271Z aac Set GS Fay 2S ——— Fae eon ht Seah el a Pa, Tm gitah es = 271Z 24,2 Sinh* (==) Fpy = Ost L 271d 2 eR 2T sinh Gia) 126 cos 9 sin 0 [cos* (6 - 6) - k] |sin 6| (AS) (A6) (A7) (A8) 2 271Z BANZH? SCCS S| ome Fly = 2 3 5 otd ° (A9) Dl sinh Cr The expressions, Fy, Foy? and Fry, are constant for a given set of wave and pipeline conditions. Linear wave theory was used in the analysis because, as discussed previously, there seems to be no obvious way of accurately describing the lift force phenomenon mathematically using higher order theories. Since the iift forces are much larger than the vertical drag or inertial forces, with the drag forces being almost completely insignificant, there was no point in using higher theories to express the vertical components of the drag and inertial forces. For any vertical wave force record in which the corresponding wave and pipeline conditions are known, a least squares analysis can be performed on the data to determine the values of the unknown parameters Cy, >, k, Cy, and Cp in the vertical wave force equation. The least Squares analysis yields the values of these five parameters which best fit the force data throughout the entire wave cycle. This is accomplished by determining the values of these parameters which minimizes the sum of squares of the difference between the observed force data and the corre- sponding forces calculated with the mathematical model throughout a complete wave cycle. Using the appropriate trigonometric identities, cosa) (Olle ¢) = 1/2 + 1/2 cos 2 (8) = o) IW/2 > l/2 (COS*ZE COs ZW PB Siim BP sim Zo), (A10) so the lift force equation can be expressed as: Fy, = 1/2C; pAu 1/2cos 26 cos 28+ 1/2sin 26 sin 20 + 1/2 = k] (All) 2 max [ Or Fy = Ay CoS Ae eh) Si 20) Cy (A12) where A, = 1/4 Cy P A upg, cos 26 = 1/2 Cy Fy, cos 26 (A13) Bil 1/4 (Cp ow AC WEES Us iin s20"= MI/2 CAD, Sam 20 (A14) 127 = 2 = = = Ce WANG oN Gye = 1) 26 mB (U2 = i) (A15) In an analogous manner, the vertical components of the inertial and drag forces can be written as: CDE = Gy @ Gale cos 0 = D, cos 0 (A16) and (Fp) = 1/2C) pA v_. sin 6 [wanes sin 8| = E, sin @ |sin 6], (A17) where Dy = Cu 0 es =-C Pu (A18) t max M IV : 271Z In (=) Ov Rhee sie eee re) 2d (A19) max lk sinh (aE? By US Gy DN We (eel = Sy By (A209) : 211Z mie aa iia ‘ hy SIE en aR on (A21) max T f 271d sinh (Ea The total vertical wave force at any position 05 in the wave cycle can then be written as: Be cS) = Fe + ee + Ce = AY cos 26, + B. sin 20, & 1 I + D_ cos 05 + E sin 0. |sin 6. |. (A22) 1 The parameters Aj, B,, Cj, Dy, and E, are constant for any given values of Cy, >, k, Gy, and Cp, corresponding to the particular wave and pipe- line conditions under consideration. The sum of squares of the differences between the observed vertical forces, Bo yeeé and the corresponding calculated forces, F (85); is written as: 128 n 2 pa . UR (9; ) - BERG) = 2 [A, cos 26. + By sin 205 + C, + Dy cos 0. lows — . . = 2 gost @), |sin6, | ne ry lige (A23) To minimize the sum of squares of the differences, the derivative of this expression is taken separately with respect to each of the five unknown parameters A,, B,, C,, D,, and E,, and the resulting expressions are set equal to zero, yielding a system of five simultaneous equations with five unknowns. The system of equations is then summed for each interval, i, over a complete wave cycle, and the resulting expressions are solved for the values of the unknown parameters A,, B,, C)> D> and Ey which thus minimize the sum of squares of the differences. The derivatives are: [ps = BGs oA = 2A cos* 20. + 2B l 1 1 Sim 262 Gos 2), 1 il 1 2 AC. COS ZO, + 2D. cos 8, Cos 2G. il 1 it TL 1 B22 sln@, |Sin ©.) cos 20. 1 st al 1 = 2) en ORD cos 26. = © (A24) ed. = Fed" = 5 2 5 2 3B, 2A, cos 20. sin 20. + Bysin 20. - AG Sin 2, = Ad Cos Oy Sim EO. 1 1 1 1 1 = De sim 6, ||siin @. || sim 20. 1 aL 1 1 - 2F (8, ) sin 26. = @ (A25) ate ee) = 8.8.) 7G = 2A, cos 20. + 2B.sin 20. il 1 + 2C, + 2D, cos Fy = Ascii GQ. crn a] It 1 1 = 28 (On) =O (A26) 129 eC) CII oD) OU (Ga) > Bx oe Ie dE, = Pi. Cos Za, Cos, @, 1 1 1 =) 23 sion A), Cos O, + AG_cos @. i 1 1 1 1 2 ap D Dj cos oF + 2E - 2F (8, ) cos 0, = @ = Du cos 20, sim O. sim @, | 1 a a 1 oo 2B sin 26, sim 8, || siti @, || 1 a 1 1 2 2C,Siim @, (Sim ©, | 1 1 i + 2Q0.cos ©. sim @, |sim ©. | 1 a 1 1 g o 9) tp 2E, (sin Oo, |sin 6.1) = 28 (6.) sim @, |sim @.| | © OW ~ oi i i sim @, ||sim @.|| CoS 2. iL i iL (A27) (A28) For an even number of equally spaced time intervals, 8., summed over a complete wave cycle, many of the terms cancel out due to the symmetry of these sinusoidal functions. Thus, COS COs sin cos 8. = 0 i 29), = 0 il 20, =O il Oy sim Os) =O i Oo. cos 2, = O i i 130 (A29) (A30) (A31) (A32) (A33) n ) cos @, sam @. sim 6,|| = © : i i i n ) COS 48), sim Zea. S © a a, ait I (jo) n ) cos 20. sin 9. |sin 8. | = al i i I (=) nN ) sin 26, sin ©. [sim 9, | = 1 1 1 (A34) (A35) (A36) (A37) (A38) when taken over a complete wave cycle, As a result, only the squared terms, and the terms involving the observed forces, F, (6;), remain in these equations. The resulting expressions are: n n 2 A ) cos 26. - 2 F iy 695) cos 20. Tl et b=) 29] ° < ~~ D fh C9 n h 3 NO fap) | ed I 131 (A39) (A40) (A41) (A42) n ED) (sin 9. |sin O21) 1=1 n = L See )) Sei |sin 8. | = 0) (A43) where n is the total number of values taken from the vertical wave force record (from an even number of equally spaced intervals per wave cycle, and over any number of complete wave cycles), and i is the number of the interval. These expressions are easily solved for the unknown parameters A> BL; C)> D)> and Ey» yielding: n ) 2 (@a) Cos 20. es OV 1 A= Se (A44) ) cos* 20, i=l n ) BF 0@.)) sim 26. 51 OV at 1 We os ea OU GRE =e (A45) ) sin? 20. i=l 2 n os ov Oi? C, = = (A460) n 2 Been) GOS Oe 0, = ae (A47) cos Oo; i=l 132 n ) Be (Cey) sin 05 |sin 0. | E, = ______—_ (A48) c C0 2 i (sin 0; |sin 8-1) With these relationships, the corresponding values of the parameters C;, >, k, Cy, and Cp in the vertical wave force equation which best fit the data throughout the complete wave cycle can be obtained. The coefficients of mass and drag, Cy and Cy» are obtained directly from the parameters D, and Ey: since 1 n D, 4 Oa) Cos Oe va My ; Fs , cos? 6 ae Mv i=l 1 a : 9 z, & a @-)) Sin Gist | : “Dv F y (sin 9, |sin 0.|)? ae Dv eet i 1 Since Ay = 1/2 C, Bie cos 26 and B, = 1/2 Gi Piigy sin 2p, the phase shift parameter > can be obtained from: B & 2 1/2 (os) = 1/2 eee: any = 2 ean ee (A51) B since 1/2 C, F cancels out of the expression +). Thus, Ib, Jin g = 1/2 tan (A52) 133 After » is known, the coefficient of lift, C;, can be obtained from Ghhidnere AG) Cie Bi5 Sse Ay hae 1=1 ous 1/2 Fi, cos 26 n (S3) 1/2 Fy, cos 29 2) cos” 20. i=] n By al a, (A), Sian. Ae l= of Cl 5 /2mE usa n Coe 2 i, Sm 2H) sim 2s i=1 Alternatively, Cz, could be obtained from A, and B, directly without first solving for $, since eS Bil G2 GC, ie os 20)" = C2 GC, Bia etm 20)° = V(@Y/2 CG, Bra” (Gos> Bo = sim” 20) = 2 G, Pay 3 (AS5) Bar Oa) Sit 20, n cos* 20; } sim 20. hE Gg (A56) 134 Finally, the parameter k can be obtained from C, knowing the value of C,, ,since n Gi a Pov ey 1= n Nee 2 ES ND) ee (A57) Cr OP Ly Cr FLy Thus, once the vertical wave forces on a pipeline are measured experimentally, the values of the parameters Ch Oey ke Cy, and Ch of the vertical wave force equation which best fit the data throughout the entire wave cycle can be determined for the particular set of wave and pipeline conditions tested. In an analogous manner, the least squares analysis car be applied to the horizontal wave force data. Omitting the horizontsl force associated with eddy shedding, the horizontal component of the wave- induced force can be expressed as equation (2): p A, = Eo, ? Goy = Gyo Vase 72 Go halal. (2) The data from the horizontal force measurements show that the horizontal eddy forces are insignificant in comparison to the horizontal drag and inertial forces for the experimental conditions tested. Using linear wave theory, the horizontal components of the wave kinematics with respect to time can be expressed as: 2712 fe TH cosh (a 4 ur Tra. oS Sima C= I 271Z 2,, cosh (—— au 2teu = 5 sin 6. (AS8) tS sini =) ISS Substituting these expressions into the horizontal wave force equation yields: 27 oAn2H2 cosh? eA Re i (C = cos 6 |cos 6| h D A D) 271d 2T sinh Gime! 277Zz 2 cosh (CG) -Cy even Z te sin 6 (A59) i 1d W sinh (——) Ib or F, = Cp Fp, Cos Is) |cos 6| - Cy Fy, sin 6 (A60) where 27Z oAn?H2 cosh’ (=) PD ety 2 Gee OTE Ge) Dp sinh? Ga 21Z 2 cosh (——) rye seem L (A62) T sinh (ATS The expressions Fp, and Fy, are constant for a given set of wave and pipeline conditions. The horizontal component of the wave-induced force can also be written as: ht = Ay COS © |cos6| + By sin 0 (A63) where MS Gy ap, = 2 Gy OA wie peel (A64) 136 27Z _ oH cosh Gime) Clee ape oe aad (A65) sinh (——) L Bae: Crease ou 2 > ~ Ga Pun = “u © VY GD (A66) max 271Z du men ost Ge) Gz 25 Se SS Sse (A67) dt F 21d max ih sinh Cores! Thus, the total horizontal wave force at any position 0; in the wave eyelel can bieexpresisied asi: CO) = Cpe = Cie = A, cos 8; |cos 8; | BD Sumi oF, (A68) where the parameters A, and B2 are constant for any given values of Cy and Cy, corresponding to the particular wave and pipeline conditions under consideration. The sum of squares of the differences between the observed horizontal forces, Fo, (81), and the corresponding calculated forces, Fy (@5)5 2s written as: [#, ©.) = By, @adle = [Ay cos 6; |cos 6, | I i=l lors los 7 By ci On = 254 (aioe (A69) The derivatives of this expression taken with respect to the unknown parameters Az and B2 and set equal to zero give the following equations: DR, (Ox) = Bay (Onde dA, = 2 A, (cos 8} |cos 6, | 2 137 + 2B, sin 8. cos 6, |cos 0. | 1 i i - 2 a (8. ) cos 0. [cos 8. | = 0 (A70) Ai (Gee uss ONS Oe i ona) DN cos @, |G || sim 8, oB., 2 1 1 i = 0. (A71) n Simee |) ) stim o5 cos 0, |cos 0. | = 0 for an even number of equally i=1 spaced intervals 0. summed over a complete wave cycle, the resulting summed expressions for the derivatives set equal to zero are n n A, 1) (cos oy |cos chp - » Bh (98; ) cos 0, |cos 8. | = 0 (A72) 1=] 1=1 n n . 2) a a B, 4 sin 0, - ?) Bas (9. ) sin 0. = 0. (A73) = i=l These expressions are easily solved for the unknown parameters A, and Bo» yielding: n ) =1 Aoi iqenT Hliw NET Dae caa Ty: (A74) 2 ye (cos 05 [cos 8.1) i=l ae. (8, ) cos 0, [cos 8. | 138 B. = eel be TS REM A (A75) The coefficients of mass and drag which best fit the horizontal wave force data throughout the entire wave cycle can thus be obtained directly from the parameters A, and BA since n A, J Fe (8. ) cos Oo. [cos 8. | Ch Sm (A76) Dh = 2 Poh } (cos 05 |cos 8, |) alt n é J By, i.) sta 8, chai 202 —_—_ .. (A77) Mestape : h. S > Mh Fuh ) sim © ISg) APPENDIX B COMPUTER PROGRAM FOR VERTICAL LEAST SQUARES ANALYSIS (TWO-DIMENSIONAL DATA) 140 PROGRAM SML OIMENSION X (41),F PC 41) FF C41)» 1Fv(41) ,RES( )) SET TEST CONDI ANGLE =00 O=20e000 FEAC IN DIGITIZED DATA 8 FEAD 1,»UP,DN UF »DF ,OIA, XC IF (UP )1091099 9 READ 2_pCh pT yNo_ XWoXF pC yg WOoF Og (HI CI), T=1 94) FEAD 3,°¢FI (1) »T=1_940) 1 FORMAT ( 2F 30 35 2F 300» 2F 303) 2 FORMAT (F3e39F 3029129 2F 201, 7F 300) 3 FORMAT(24F3e0) DETERMINE wA4VE HEIGHT OO 11 IT=1y% IF CHICT)—-wO)21523,23 21 HRC T) =-CWO-HI (1) )*®CONZC)¥(1 o/ XW) GO TO ll 23 HX( 1) =(HIC IT )—-wO) *(UP/SC)*(C Lo XW) 21 CONTIN JE He (HXC 1) ¢HX02)-HX(3)-HX( 4) ) 720 CALCULATE CONSTANTS IN FORCE EQUATION PLl=3e Ll F41S92EE 36 R=12938 CALL wWAVEL(T,D,XtL) ZV2=CL¢(eS*OLA) A=EXP((20e¥*PI*ZV)/XL) COSHA=A+4(1le ZA) SINHA=A4—-(10/A) B=EXP( (2e*PI*0)/xXL) SINHB=B8—-( 12/78) CS=COSHA/SINHB SS=SINHA/SI NHB CF LSR¥OLAEXCHH*eP IFO L/S ( 20% T¥T) FULV=CF1*H*®CS*CS FOV=CF1*H*SS*SS FMVSCFI¥*OLA*PI¥*SS U=(H*CS*PI )/T DETERMINE AVERAGE, MAXIMUM, AND MINIMUM FORCES FMAX=—-05 170974971 )-FO)*(DF/C) #000 2204/7KF Y-FO)*(UF/C) 0 002204/ XF eFMAX)902,901 eFMIN)903912 l2 SFs=SF¢FP 13 CONTINUE ‘ SF=SF/400 14 | OT =e 31415926536 Az-DT DO 15 1=1,40 Az=A*DT XC(I)3COS (A) B=2e*A v(I)=COS(B) G(I)2=SIN(B) C=SIN(A) Z(1)2C*ABS (C) FC I)3FP(1)-SF SFFzSFFeF(I)® SF X=SFX¢F(I SFY=SFY+tF (I SFQ=SFQ¢F(I SF ZsSFZtF (1 1S CCNTINUE AXZSFX/200 AV=SFY/20. AQ=SFQ/206. AZ=SFZ/156 VX=AXK*E SF X VY =AY*5SFY VQ=AQ*SFQ VZZAZ*SFZ VR 2SFF-VxX-VY-vZz-va Cc CALCULATE COEFFICIENTS AND PARAMETERS PHI AND K PHI 3228e64789% ATAN2( AQ,AY?) IF (PHI ol Te-450 )7999,8999 7999 PHI=PHI 4180. 8999 CONTINUE YA=SQRT (AQ* AQ+t AY¥AY ) CLV=2e *¥VYA/FLV ANG=(ANGLE*PI )/1800 CLVAZCLV/COS( ANG) CLVU=CLVA/COS( ANG) CMV=-AX/FMV COv=-AZ/FOvV XK=0e5-(SF/(CLV*FLV) ) cS PRINT RESULTS OF ANALYSIS PRINT 200 200 FCRMAT(1HI1) PRINT 4 & FORMAT (10X_,6HT (SEC) » 7X, 6HHT (FT) 9 SX_9 SHWAVEL (FT) «6X, 7HOEP(FT) »4X,9HU IMAX( FPS) 94X_9HCLEAR(FT) 9 6X_ THDIAC FT), 3X9 12HCYL LGTH( FT), &X» BHANG(O 2EG)) PRINT 3009T pHyo XL »D_,U,CL »DI Ag XC, ANGLE 300 FORMAT (2X 5F 130 29F lL 3e3pFl Ze2oSF lise 39FlZel////) PRINT 307 307 FORMAT(6X,12HTOTAL SUM S0,6X_5HCOS2A 91 OX ep SHSIN2ZAG 11K, @HCOSAy BX, LOH 3 it oe NOFICT) » TEl , 40) 1 FORMAT (2F 30 3) 2F 300) 2 FORMAT (F303 )F 302912, 2F201,I3F 300) 3 FORMAT ( 24F 3.0) C OETERMINE WAVE HEIGHT DO 11 LT=i531,10 IF (VICI )-WO)21, 23,23 2. YS(1)=-(BO-VI(L) DECONZC RCL os XW) GO TO iit 23 YSCL)=(YICI)-wO)*(UP/C)*( les XW) 14 CONTINUE He(VS(1)¢VS(21)-VS(211)-VS(31) D720 C CALCULATE CONSTANTS IN FORCE EQUATION PI23e141E92ES 36 F2z12e938 CALL WAVEL(T,0O,XL) ZV32CL¢(e5¥*OTA) A=EXP((2e*PI*ZV)/XL) COSHA=A+(1e ZA) BrEXP((2e*P1*0)/XL) SI NHB=B— (le /B) CS2COSHAZSINHB CF LSR*eDIAXXCRHSPLEPIsS(20¥*T*T) FOH=CF 1*H*CS*CS FMH2ZCF1*OLA*PI*Cs Us(H*CS*PI )/T Cc OE TERMINE AVERAGE? MAXIMUM, AND MINIMUM FORCES FMAX=—-05 UT »,OUTPUT ,PUNCH Dy iestal bev ttsa ds FICO)» FPC 41) oF C41) FHI 41), =a 2uvU 240 FO) 701,702,702 =F ICL) )®(CED/C) ®0002208/XF 13 CONTINUE SF2SF/406 is CALCULATE SUMS OF SQUARES AND PRODUCTS 150 SFF=00 SFP=Q. SFZ=006 OT =. 31415926536 Az=-DT OO 1S L=1,4a AZA+0T P(LT)=SINGA) Cc=COs(A) Z(1)=C*ABS (C) F(I)sFPCI) SFF=SFFCF (1 )*F C(I) SFP=SFPCF (I )*PC(T) SFZ=SFZ+F(1)*Z¢1) 1S CONTINUE AP=SFP/206¢ AZ=SFZ/15e VP=AP#SFEP_ VZ=AZ*SFZ Cc CALCULATE COEFFICIENTS CMH=-AP/FMH COH2AZ/F DH Cc PRINT RESULTS OF ANALYSIS FRINT 200 200 FORMAT(1H1) PRINT 4 i Eee 4 FORMAT(10X, 6HT (SEC) 9 7X» 6HHT (FT) 9 SK sOHWAVEL (FT) ) 6X4 7THOEP (ET) » 4X) SHU LMAX(FPS) » 4X) SHCLEAR CFT} » 6X» THOIACE TT)» 3X, L2HCVYL LGTH(FT) » 4X», BHANG(D — PRINT 300,T oH, Xi »pDoUp,CiL -.DIA,XC,ANGLE 300 BOR MATIC2XG Cl Sei2) Gll3¢ 39h Seay Sr ilse 3501 3= 17777) N 330 FORMAT(26X,12HTOTAL SUM SQ_7X_ SHSIN», 10%, 8HCOS/COS/) PRINT 331eSFFeVP eVvVZ 331 FORMAT (20X», 3F 1506/7) PRINT 332 332 FORMAT (46X_ 2ZHAP 7 13X 9 2HAZ) PRINT 333,AP,A 333 FORMAT (35X )2F1506/) PRINT 334 334 FORMAT (46x SEE Arg h2 Mie SCO) PRINT 335eFMHeFOH 335 FORMAT ( 35X» 2F1506////) FRINT 336 336 FORMAT (46X_ 3HCMH» 12 » SHCOH) PRINT 337¢ CMH, CDH 337 FORMAT(35X,2F1ISe3////) PRINT 309 309 FORMAT (38X, BHF AVG(LB) 9 7X» BHF MAX(LB) » 7X, BHEMIN(LB) ) PRINT 304,SF pFMAXoFMIN 304 FORMAT (30X, FlSe62F 15e57777) PUNCH 387,CL,DIA, ANGLE T 9H» XL gp U_y CMH, COOH, SF 387 FORMAT (F 4s 35F S03 oF 400,f502¢F So 3,F602,F 604, 2F802,F 1006) Cc OT le DATA AND RESULTS FOR COMPARISON N 26 FCRMAT(4X,7H FP(LB), 3X9 7H FH( LB), 2X,8H RES(LB)/) 15| 31 100 32 10 Cc 2 BB=35e/8 ©O 31 L=1,101 G(L)2=3H CONTINUE 00 32 I=1 FHC 1) @COMSFON#Z(1)-CMH®EMHEPLT) RES(L) sFP(T)-FH(E) G(51)=1HI AAODana CCNTINUE END Eee rae Lets ne SUBROUTINE WAVEL(T)D, XL) B23202*T*T/60 283185 TPO0=260 2831 85D IF(B-TPD) 22253 OEEP WATER INITIAL ESTIMATE FOR WAVELENGTH xL 2B GO TO 4 Cc es INITIAL ESTIMATE FOR WAVELENGTH * Ss XL =T*SQRT (O0*3202) XU X= XL XL 2B TANH( TPO/XLX) IF (ABS (XLX—-XL)— 0005 55454 RE TURN END 152 APPENDIX E TABULATED VERTICAL FORCE DATA FROM TWO-DIMENSIONAL EXPERIMENTS 153 CLER DIA ANG e001 0001 pep pe eer eeceoeoeroesre2e00 0°80 (-Yo¥ NK ~~ K Nok K-10 0-1-0) NT OT ee ry GAaecacecesco NaN epCoepeceeseeeroeeHaFO2S2F2022SR2SGC9R8F08980088 ee ee ee ee ee ee ee eee eee ee ADBHAKRAFHOAOCAKHAGHOAAPCAGCAORBECAG NNN NSIS ISIN SSS ° ~ co @Q 9256 4307 9elT7 96 2b UMAK o294i 04423 06808 24063 05350 05756 03325 04898 05823 02676 04341 oS774 22902 04148 06929 04180 25280 of277 05402 eb6CT6 02816 04457 05730 03014 02882 e 391C 06766 e4172 05429 06668 03705 05336 26119 ©2944 04579 03053 05701 04174 S727 04033 0 4465 68389 civ 5054 4043 3017 4059 3045 3050 4045 3070 3006 4014 3033 2268 7206 Ge 86 4037 6e57 Seal 4039 4091 3087 £208 4020 3009 2057 2068 $006 4.59 6002 SeSi 4007 6014 Sel2 4200 5e39 4e24 3e37 4019 S072 4.08 4078 4028 3099 4eS7 40f3 3068 4022 3088 3017 6el3 6047 6092 6089 7033 6060 6008 7042 6060 €.08 6020 S046 407k 30 55 6009 6046 Se32. 5093 6024 $094 6030. 5.99 5096 4096 Se28 4042 4036 4o13 6035 4034 3088 3062 4027 CLVA Be54 4043 Bol? 4059 3045 3050 4045 3070 3006 401% 3038 2088 To 06 6086 4eS7 6057 Se4l 4039 4091 3087 6208 4020 3009 2057 2068 5e06 4059 6002 Sef 4eC7 6ol% Sel2 4.00 3.59 4024 3037 4019 Se72 42086 40.78 4026 | 3099 4057 4o.63 3068 %o22 3e6) 3oi7? 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4064 403% 4034 4e73 4073 5050 5050 S$e60 5060 5045 5045S Se79 Se79 3092 3092 4038 4038 4083 4083 S02) Se2k 6e42 6042 5068 5068 6e25 6028 6eS1 60S) 6085 6085 3099 3099 4077 4o77 4o7L 4073 Se18 Sei8 6054 6084 Se70 Se70 5e97T S097 2017 2017 2e31 2031 2023 20023 2267 2067 2081 2081 2075 2076 2066 2066 2073 2073 2082 2082 2095 2098 3e21 3e21 3008 3008 2e71 2e71 2ell 2e11 2067 2067 3027 3027 3056 3056 3099 3099 Geo57 4057 Se08 5008 2061 3030 3034 3049 3067 3067 3038 3088 4054 4054 4095 4095 4099 4099 Se Ol SeOl 6003 6003 2094 2094 3e4S 3045 4026 4026 Se07 SeC7 Se35 5035 Se4S Se4S Se13 5013 156 cov 47095 39040 56007 52066 89019 86070 118013 139026 115069 206049 257036 133064 -117043 64028 -37028 -18019 24075 —26025 21066 -17081 -19009 6030 17033 -140e71 -1560e24 -58016 -68206 47038 - 32074 22042 -21e78 -3003 -258e32 -213097 —tlledSl -54027 -65e91 48036 “10099 -22.068 -e75 -85e76 42007 - 392299 -134e30 -112064 -29e39 12e1l 68282 58039 CLER 2 OES 2083 e 063 e083 e 083 2 0863 e083 2 083 20083 20 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03770 18091 00786 18074 01402 18974 92028 19017 02692 18083 e313 19000 .3€38 19000 03936 157 CLVA 050 048 of3 0°66 058 058 0€2 lei3Z 094 082 o78 076 280 °70 °88 12018 092 e799 1eO01 097 097 098 1010 eS6 1003 1030 047 097 099 1000 1018 1e13 te3l 064 023 e2€ 029 022 e2) 036 094 e7S eS 056 050 057 e41 081 027 090 074 e65 064 o71 039 050 o73 053 089 1207 1202 094 06S 0&2 CLVU 060 048 063 2°66 058 eS8 062 lel3 2094 0°82 078 076 280 070 288 1613 092 °79 1201 097 e97 038 1010 096 1003 1¢30 $47 o97 099 1200 1018 1o13 le3l 064 223 026 e029 022 0°29 036 09% e7S oS 055 eS0 0S7 041 eAl 027 290 074 065 064 e7l 089 250 e773 ef3 2°89 1007 1202 094 085 082 DUMNUABDONWVONODO PKOKFUFNORADOWAD=— VUE ee - 42043 APPENDIX F TABULATED VERTICAL FORCE DATA FROM THREE-DIMENSIONAL EXPERIMENTS 158 CLER 0001 e001 e001 2001 20001 2001 e001 ecOl 0001 eCcol ecol eCOl eV01l 2001 2001 20001 e001 e001 ©9001 e001 e001 2001 e001 0001 e001 2e001 2001 e001 e001 eCOl e000) e001 2001 eCOl 2001 2001 2001 e001 e001 e001 e001 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