WAVE FORCES AND MOMENTS ON HEAVILY OVERTOPPED VERTICAL CURRENT DEFLECTION DIKE

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1 006, Proceedings of te First International Conference on te Application of Pysical Modelling to Port and Coastal Protection. ISBN xxx-xxxx-xx-x WAVE FORCES AND MOMENTS ON HEAVILY OVERTOPPED VERTICAL CURRENT DEFLECTION DIKE STEVEN A. HUGHES (1), HARLEY S. WINER () & EDWARD R. BLODGETT (3) (1) Senior Researc Engineer, Coastal and Hydraulics Laboratory, U.S. Army Corps of Engineers 3909 Halls Ferry Road, Vicksburg, MS 39180, USA. () Cief, Coastal Engineering Section, New Orleans District, U.S. Army Corps of Engineers, 7400 Leake Avenue, New Orleans,LA 70118, USA. (3) Hydraulic Engineer, Coastal Engineering Section, New Orleans District, U.S. Army Corps of Engineers, 7400 Leake Avenue, New Orleans, LA 70118, USA. Abstract Irregular wave forces on a eavily overtopped tinned-wall vertical current deflection dike were measured during a series of laboratory experiments. Te purpose of te experiments was to obtain site-specific engineering values for use in design of a dike located at te mout of te Mississippi River and to develop generic design guidance for eavily overtopped vertical walls. Predictive equations for te root-mean-squared soreward-directed peak total force and associated moment arm were developed in terms of te incident wave momentum flux and relative wall eigt. 1. Introduction and Background Te focus of tis study was a vertical, tin-wall dike located at te mout of te Soutwest Pass cannel of te Mississippi River in Louisiana, USA. Te dike extends perpendicular from te west river jetty approximately 45 m (800 ft) into te river as sown on Figure 1, and its function is to divert river flow toward te main navigation cannel to elp prevent cannel soaling. Water depts along te dike range up to 6.4 m (1 ft). Te most recent wooden dike structure suffered extensive wave damage annually, and it is to be replaced wit a new steel vertical wall aving a top elevation eiter near or below Gulf mean lower low water (mllw) level. Tus, during storms te wall will be eavily overtopped, wit te upper portion of storm waves passing over te wall. River currents in te upper few meters of te water column will not be diverted by te dike, but instead will continue to flow seaward over te wall and act as an opposing current on te incoming wave crests. Te objectives of tis study were (1) to determine specific force and moment loading on te wall for design of tis specific structure, and () to develop generic design guidance applicable for similar situations. Placing te top of te dike at an elevation comparable to te incoming wave troug introduces several ydrodynamic complexities. Te dike will be eavily overtopped during storm conditions, and te overflowing water will cause a region of flow separation and lower pressure on te leeside of te wall (Knott and Mackley 1980). Tis low pressure will increase te soreward-directed wave force at te top of te wall. Incoming waveform caracteristics 1

2 Report Documentation Page Form Approved OMB No Public reporting burden for te collection of information is estimated to average 1 our per response, including te time for reviewing instructions, searcing existing data sources, gatering and maintaining te data needed, and completing and reviewing te collection of information. Send comments regarding tis burden estimate or any oter aspect of tis collection of information, including suggestions for reducing tis burden, to Wasington Headquarters Services, Directorate for Information Operations and Reports, 115 Jefferson Davis Higway, Suite 104, Arlington VA Respondents sould be aware tat notwitstanding any oter provision of law, no person sall be subject to a penalty for failing to comply wit a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 006. REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Wave Forces and Moments on Heavily Overtopped Vertical Current Deflection Dike 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) U.S. Army Corps of Engineers,Coastal and Hydraulics Laboratory,3909 Halls Ferry Road,Vicksburg,MS, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES Proceedings of te First International Conference on te Application of Pysical Modelling to Port and Coastal Protection, 8-10 May 006, Porto, Portugal 14. ABSTRACT Irregular wave forces on a eavily overtopped tinned-wall vertical current deflection dike were measured during a series of laboratory experiments. Te purpose of te experiments was to obtain site-specific engineering values for use in design of a dike located at te mout of te Mississippi River and to develop generic design guidance for eavily overtopped vertical walls. Predictive equations for te root-mean-squared soreward-directed peak total force and associated moment arm were developed in terms of te incident wave momentum flux and relative wall eigt. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Same as Report (SAR) 18. NUMBER OF PAGES 10 19a. NAME OF RESPONSIBLE PERSON Standard Form 98 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 will be altered by bot te partial wave reflection at te wall, and by te out-flowing river current tat will steepen te incident wave crests to some degree. Figure 1. Current deflection dike location at te mout of te Mississippi River. Experiments, Measurements, and Analyses.1 Experiment Parameters. Experiments were conducted in a large basin at a geometrically undistorted lengt scale of 1:50 wit ydrodynamics and measured forces scaled according to Froude similarity. Tis lengt scale was cosen to best suit an existing experimental setup in te tidal inlet simulation facility at te Coastal and Hydraulics Laboratory in Vicksburg, Mississippi, USA. Te vertical dike was situated in te basin similar to sown in Figure 1. However, in te model te east jetty did not extend as far seaward as in te prototype. Te key parameters for te experiments (in prototype-scale units) are as follows. Water elevation was eld constant at m mllw and te seafloor elevation at te wall was 6.7 m mllw giving a total water dept of 7.5 m for all experiments. Offsore dept at te plungertype wavemaker was 15.8 m. Four wall eigts were tested aving top elevations of m, m, +0 m, and m mllw. Top wall elevations relative to still water level were.38 m, m, m, and m, respectively. Irregular wave conditions were near dept-limited breaking at te wall for many of te tests. Zerot-moment significant wave eigts (H mo ) varied between 1.5 and 3.7 m, and te wave period associated wit te peak of te wave spectrum (T p ) varied between 7.0 and 13.5 sec. Experiments were conducted witout current and wit seaward-flowing average current of 0.9 and 1.8 m/sec (prototype-scale units).. Measurements. Te key measurements of tese experiments were te incoming waves and te resultant forces on te overtopped vertical wall. Capacitance wave gauges recorded variations in te water surface elevation at a 0-Hz rate trougout te entire 360-sec duration of eac experiment. Number of waves per experiment varied between 190 and 360 waves, depending on wave period. Wave data were collected near te 4-m-long wavemaker, at a tree-gauge array just seaward of te vertical dike, soreward of te vertical dike, and at a gauge on te same dept contour as te array but far enoug to te side to be free of waves reflected by te wall. Tis latter gauge provided te incident wave parameter estimates. Frequency domain wave parameters were used in te analyses presented in tis paper. Wave forces on te vertical dike were measured using te apparatus sown in te center potograp in Figure. Tis force-measuring portion is te cantilevered wall section supported by te vertical framework. Narrow gaps separate te supported wall section from

4 te adjacent fixed wall. Wave forces applied over te wall section result in reactions at te upper supports as illustrated by te free-body diagram in Figure. Two force transducers were used at te fulcrum point (F and F 3 ) and a tird transducer was used at te top (F 1 ). Analysis of te free-body diagram at any time yields te total wave force F L and te corresponding moment arm L F about te wall base at tat instant. Figure. Force measuring section of model vertical dike Te force transducer cells were calibrated over te range of expected loading using pure axial loading. At te start and end of eac day of testing te force-measuring device calibration was cecked in place by affixing a calibration jig, as sown in te rigt-and poto of Figure, and applying a series of known loads in te tray. Tis exerted a orizontal load on te wall section of known force and moment arm from wic forces on te load cells could be cecked. (Calibration cecking was done wit water in te model to assure any buoyancy forces were included.) Calibration loads were applied in bot te soreward and seaward directions. Te calibration ceck revealed a small residual resistance moment in te force-measuring section caused by rigid connections between te supports and te load cells. Te free-body analysis assumes frictionless pin connections. Calibration measurements were used to quantify te residual moment as an empirical equation over te range of loadings so tat it could be removed from experiment force measurements. Prior to te start of waves for eac experiment te load cells were zeroed to sow no force on te wall. For tose experiments wit constant seaward current, te current was allowed to reac a quasi-steady flow state before te force transducers were set to zero. Tus, forces recorded for eac experiment represent te loading due solely to wave motions. After resetting te force transducers, waves were generated in te model; and sortly after te first waves reaced te vertical wall, force data collection was started. Force data were collected from te tree load cells at a 40-Hz rate for te same 360-second time period as te wave data. Tis logging rate was sufficient for recording pulsating wave loads, but te rate was not ig enoug to record impact loads. Because te top elevation of te wall was well beneat te wave crest elevation, impact loading was muc less probable tan for emergent walls. 3

5 .3 Force Data Analysis. For eac experiment te tree syncronous force time series were first converted to engineering units (grams) and ten combined at eac time step according to te force balance equations derived from te free-body diagram of Figure. Tis resulted in time series of te total force on te force-measuring section (F L ) and te corresponding moment arm about te seabed (L F ). Te residual moment was included in te moment arm calculation. Model force values were divided by te orizontal widt of te force section to give force per unit lengt, and appropriate scale and conversion factors were applied to yield prototype-scale force per unit lengt in Englis units. Te force time series exibited caracteristic sarp peaks in te soreward direction (defined as negative) corresponding to te wave crests, and broad lower peaks in te seaward direction (positive) resulting from te passage of te wave troug over te wall. Design of eavily overtopped vertical walls requires estimates of te largest forces and moments for a specified wave condition. For eac time series te soreward-directed and seaward-directed peak forces were extracted and plotted as distributions normalized by te root-mean-squared of te peak forces (F rms ) for te time series. Similarly, te distributions of soreward- and seaward-directed peak moments were determined as te product of peak force and corresponding moment arm. Summary statistics from time domain analysis of te soreward- and seaward-directed peak forces and moments were tabulated, plotted, and inspected for obvious trends. Forces and moments were always larger in te soreward direction, and magnitudes increased wit iger wall top elevation, iger zerot-moment wave eigt, and longer peak wave periods. Te effect of te seaward-directed current only caused a minor reduction in wave force on te vertical wall, and te current effect is not discussed furter in tis paper. However, te current does exert a seaward-directed steady force on te vertical wall tat needs to be considered in design. 3. Force Distributions Figure 3 sows typical results of te force and moment distributions. Te upper and lower plots on te left side of Figure 3 sow te soreward- and seaward-directed normalized force distributions. Te solid curve is te Rayleig distribution based on te value of F rms for te force peak distribution. Te soreward-directed peak force distribution is well represented by te Rayleig distribution wereas te seaward-directed force distribution is a poorer matc. Tis trend was caracteristic of all te experiments. Te upper and lower plots on te rigt side of Figure 3 present te corresponding distributions of peak moment for te experiment. Te prospect tat te soreward-directed peak force distribution can be reasonably approximated by te Rayleig distribution based on te root-mean-squared peak force is a compelling concept to pursue because correlation of F rms to parameters of te incident waves will likely be more successful tan correlations based on extremes of te force distributions. Figure 4 compares actual soreward-directed peak force distribution parameters F 1/3 and F 1/10 to estimates using te Rayleig distribution based on F rms (were F 1/3 and F 1/10 are te average of te igest 1/3 and igest 1/10 of te force peaks, respectively). Estimates of F 1/3 were quite good wit little scatter around te line of equivalence. Te most variation was for te largest forces observed at te wall wit top elevation m (prototype scale) above te still water level. More scatter was seen for estimates of F 1/10 as seen on te rigt side plot in Figure 4; owever, te variation is not large, and it appears to be evenly distributed about te line of equivalence. 4

6 Figure 3. Representative distributions of soreward and seaward peak forces and moments Figure 4. Soreward-directed peak force Rayleig prediction based on measured root-mean-squared force Given te results sown in Figure 4, design guidance developed in tis study was based on estimating te soreward-directed root-mean-squared force F rms, and ten using te Rayleig distribution to estimate extreme peak forces for te wave condition. 5

7 4. RMS Force Prediction Te force parameter representing te root-mean-squared values of te soreward-directed peak forces (in Englis prototype units) is plotted versus te relative dept in Figure 5. Because te water level was te same for all experiments, te abscissa on te Figure 5 plot essentially sows only variation of peak spectral wave period. Figure 5. Root-mean-squared force versus relative water dept for all experiments Te RMS peak force increased wit increasing top wall elevation, and relative wall eigt appeared to be te most influential test parameter. Te F rms force also increased gradually as te peak spectral wave period increased (decreasing relative water dept). Most experiments were conducted wit waves approacing dept-limited breaking, and te RMS force increased as te wave eigt increased. However, experiments conducted wit lower wave eigts introduced additional scatter into te observed trends as seen in Figure 5 at relative water dept equal to about Out-flowing river currents ad a relatively minor effect on te force results, and tus, current as been neglected in tis analysis. Te next step was to normalize te RMS peak force values by an appropriate factor tat accounted for te effects of wall eigt, water dept, wave eigt and wave period. Tis factor was derived by considering te trapezoidal wave-only peak force distributions sown in Figure 6. Figure 6. Definition sketc for wave-only trapezoidal force distributions on vertical walls 6

8 Equations for te total force on te submerged wall (left side of Figure 6) and te wall extending up to te still water line (rigt side) can be written in terms of te pressures and vertical dimensions. By furter assuming te force distribution for te wall on te rigt side of Figure 6 is a linear extension of te distribution sown on te wall on te left side of te figure, it is possible to derive an expression for te force F w in terms of F o, p, and te vertical eigts, i.e., w w F w = F o + p w 1 [1] Te first term in Eq. [1] dominates for vertical walls wit top elevation near te still water level, and tis suggests tat F o / ( w /) migt be a good normalizing factor for F rms. Tere are several ways to estimate te value of F o. Tree metods were considered: F o is proportional to F o estimated from linear wave teory F o is proportional to F o estimated using Goda s (1974) metod for forces on walls F o is proportional to te total nonlinear wave momentum flux at te wave crest All tree estimates for F o in te normalizing factor gave reasonable results. However, estimates of F o based on linear wave teory (e.g., Dean and Dalrymple 1984) and calculated using te Goda metod resulted in a normalized peak RMS force tat still exibited an increasing trend wit increasing wave period. Tis is probably related to increasing wave nonlinearity tat is not captured by te first two metods. However, te wave nonlinearity was better represented wen F o was assumed to be proportional to total nonlinear wave momentum flux, M F. Huges (004) presented empirical equations to estimate te total dept-integrated wave momentum flux at te wave crest in terms of te relative wave eigt and relative water dept. For irregular waves te following formula for M F was recommended. were M F ρg A 1 = A 0 [] max gt p.06 H A 0 = mo [3] H A 1 = mo [4] and is water dept, ρ is water density, and g is gravitational acceleration. 7

9 Figure 7 presents te soreward-directed normalized root-mean-squared peak wave force using nonlinear wave momentum flux in te normalizing factor. Te relative wall eigt parameter accounts for muc of te scatter reduction, and te wave momentum flux parameter seems to ave accounted for wave nonlinearities because te data no longer exibit an increase wit wave period. Te most scatter occurs at te wave period (10 sec prototype) were additional tests were conducted wit smaller wave eigts. One reason for te scatter could be associated wit reduced flow separation as te smaller waves pass over te top of te wall. Figure 7. Soreward-directed normalized RMS peak wave force versus relative water dept Te solid orizontal line in Figure 7 was drawn as a conservative recommendation for estimating te soreward-directed RMS peak wave force acting on te overtopped vertical wall. Tis resulted in te following simple equation for estimating F rms 8 Frms w = 0.53 M F [5] were M F is calculated using te formulas given in Eq. [] [4]. Because most of te experiments were conducted wit waves approacing te dept-limiting condition, Eq. [5] may not be appropriate for smaller waves in deeper water. 5. Moment Arm Prediction For eac experiment te variation of calculated moment arm L F associated wit te peak soreward-directed wave forces was examined. Te moment arm was reasonably constant over most of te peak force range, so an average value was selected for eac experiment. Figure 8 plots te relative moment arm versus relative water dept. For reference, uniform and triangular force distributions would ave values of L F / w equal to 0.5 and 0.67, respectively. Te relative moment arm increased as te top wall elevation decreased relative to water dept indicating a nonlinear force distribution wit most of te orizontal force load

10 being applied near te top of te wall. Tis increase may be related to flow separation as te wave passed over te wall. Te relative moment arm also increased wit increasing wave period. Figure 8. Relative soreward-directed moment arm associated wit peak wave forces An appropriate normalizing factor was found by trial and error tat yielded a strictly empirical formula tat gives a conservative estimate of te moment arm as a function of water dept, wall eigt, and peak wave period, i.e., L F = 0.4 w w g T p 0.1 [6] Figure 9 sows te final normalization tat led to Eq. [6]. 6. Conclusions Wave forces on eavily overtopped vertical walls were measured for differing wall eigts and wave conditions. Te soreward-directed forces were te largest, and te data indicated te distribution of te force peaks is well represented by te Rayleig distribution based on te root-mean-squared force. An empirical equation was developed in terms of relative wall eigt and wave momentum flux to estimate te soreward-directed RMS peak force. Te associated moment arm, wic is nearly constant over most of te peak force distribution, was also given by an empirical equation. For design application, first estimate te RMS peak force, ten use te Rayleig distribution to obtain an appropriate design force (e.g., F 1/100 ), and finally estimate te moment as te product of te force and moment arm using Eq. [6]. 9

11 Figure 9. Normalized relative moment arm associated wit soreward-directed peak wave force Acknowledgments Te study described and te results presented erein, unless oterwise noted, were obtained from researc funded by te New Orleans District, US Army Corps of Engineers, and by te Coastal Inlets Researc Program at te US Army Engineer Researc and Development Center, Coastal and Hydraulics Laboratory. Permission was granted by Headquarters, U.S. Army Corps of Engineers, to publis tis information. References Dean, R. G., and Dalrymple, R. A Water Wave Mecanics for Engineers and Scientists, Prentice-Hall, Inc., Englewood Cliffs, New Jersey. ISBN Goda, Y New wave pressure formulae for composite breakwaters, Proceedings of te 14 t International Coastal Engineering Conference, Vol 3, pp Huges, S. A Wave momentum flux parameter: a descriptor for nearsore waves, Coastal Engineering, Vol 51, No. 11, pp Knott, G. F., and Mackley, M. R On eddy motions near plates and ducts induced by water waves and periodic flows, Pil. Trans. of te Royal Society, Vol 94, pp