Equilibrium Scour Depth at Tidal Inlets

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1 Coatal Enginring Tchnical Not IV-18 Equilibrium Scour Dpth at Tial Inlt by Stvn A. Hugh PURPOSE: Th Coatal Enginring Tchnical Not (CETN) hrin introuc a impl xprion rlating maximum icharg pr unit with at a location in a tial inlt to th pth of cour at that location. Application of thi proviional guianc i illutrat by thr xampl. BACKGROUND: On cour problm of concrn at improv navigation inlt occur whr th maximum pth of th quilibrium inlt throat cro ction ar ajacnt to a tabilizing jtty tructur. Th p portion along a jtty hav th potntial to unrcut th tructur to an cau ubqunt amag to th tructur armor layr. Within th inlt channl, tial currnt play a major rol in roion an poition of imnt. If cour occur clo to th awar n of th inlt, wav action an longhor currnt alo contribut to th cour action an prhap ominat th proc. Howvr, whr cour occur wll ini th ntranc channl, wav action i ruc; an it i raonabl to aum imnt movmnt at that location i rivn primarily by th tial flow. Ovr many tial cycl, cour in rgion with minimum wav action will vntually rach a liv-b quilibrium pth whr th maximum har tr acting on th bottom i no longr ufficint to initiat cour of th b. Aitional cour can occur only if th maximum flow icharg i incra at that particular location. Flow incra might occur bcau of an ovrall incra in tial prim or bcau of flow rirction rulting from tructur altration, rging activiti, or channl ralignmnt. In thi tchnical not, a nw rlationhip for u at tial inlt i vlop for th maximum tial flow icharg pr unit with a a function of th watr pth an imnt charactritic. Maurmnt of maximum icharg from Ponc Lon Inlt an Shinncock Inlt ar u to tablih an uppr-boun mpirical cofficint. Thi quilibrium icharg rlationhip impli that thr i an quilibrium pth that can tolrat a givn icharg pr unit with. Incra in icharg will rult in cour an a corrponing incra in watr pth. Practical application of thi implifi nginring approximation ar uggt. FORMULATION: Aum th vrtical vlocity profil at tim nar th maximum icharg through a tial inlt can b rprnt a a tay, fully vlop, rough, turbulnt bounary layr xtning from th bottom to th fr urfac. Any contribution by wav i nglct. Th bounary layr vlocity profil can b aquatly approximat by a 1/8 powr curv (Yalin 1971) with th har tr at th b givn a 1

2 Rport Documntation Pag Form Approv OMB No Public rporting burn for th collction of information i timat to avrag 1 hour pr rpon, incluing th tim for rviwing intruction, arching xiting ata ourc, gathring an maintaining th ata n, an complting an rviwing th collction of information. Sn commnt rgaring thi burn timat or any othr apct of thi collction of information, incluing uggtion for rucing thi burn, to Wahington Haquartr Srvic, Dirctorat for Information Opration an Rport, 115 Jffron Davi Highway, Suit 104, Arlington VA Rponnt houl b awar that notwithtaning any othr proviion of law, no pron hall b ubjct to a pnalty for failing to comply with a collction of information if it o not iplay a currntly vali OMB control numbr. 1. REPORT DATE MAR REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Equilibrium Scour Dpth at Tial Inlt (CETN IV-18) 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Hugh, Stvn A. 5. PROJECT NUMBER 5. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army Enginr Rarch an Dvlopmnt Cntr, Coatal an Hyraulic Laboratory, Vickburg, MS 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approv for public rla, itribution unlimit 13. SUPPLEMENTARY NOTES Th original ocumnt contain color imag. 11. SPONSOR/MONITOR S REPORT NUMBER(S) 14. ABSTRACT Thi Coatal Enginring Tchnical Not (CETN) introuc a impl xprion rlating maximum icharg pr unit with at a location in a tial inlt to th pth of cour at that location. Application of thi proviional guianc i illutrat by thr xampl 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT unclaifi b. ABSTRACT unclaifi c. THIS PAGE unclaifi 18. NUMBER OF PAGES 11 19a. NAME OF RESPONSIBLE PERSON Stanar Form 98 (Rv. 8-98) Prcrib by ANSI St Z39-18

3 τ V = ρ o ω C k ( h / ) 1/ 8 (1) whr ñ w V = ma nity of watr = pth-avrag vlocity C k = untrmin contant h = watr pth at maximum icharg = mian grain-iz iamtr Th contant C k i a bounary layr hap factor that inclu th unknown rlationhip btwn an bottom roughn. Th Critical Shar Str of th noncohiv an b i givn by th Shil paramtr a τ cr ( ρ ρ w ) g = C () whr C ñ = contant of proportionality = ma nity of an g = gravitational acclration = mian grain-iz iamtr For liv-b quilibrium, a har tr balanc i aum with ô o proportional to ô cr. Equating Equation 1 an rult in th xprion h = 1 ( C ) 8 ρ ρ w ρ w V g 4 (3) whr th two unknown contant, C k an C, hav bn combin into C. Th trm in quar brackt on th right-han i of Equation 3 i th ratio of grain-iz Frou numbr to th immr pcific gravity of th an, an it i fin a th Grain Mobility Numbr (Yalin 1971).

4 A mor uful form of Equation 3 i obtain by multiplying both i by h 8 an rarranging to gt an xprion for th quilibrium icharg pr unit pth, i.., q 8 9 / 8 [ g ( S 1) ] 1/ 3/ h = C (4) whr th q i fin a th Equilibrium Maximum Dicharg pr unit with, givn by q = V h (5) an S = ñ /ñ w i th imnt pcific gravity (about.65 for quartz an). A xpct, Equation 4 inicat that th quilibrium maximum icharg i primarily a function of watr pth with imnt iz having a rlativly minor ffct. MEASUREMENTS: Th unknown cofficint in Equation 4 wa mpirically valuat by compariion to fil maurmnt at two ual-jtty tial inlt. Vrtical profil of horizontal vlocity wr maur along tranct at Shinncock Inlt, Nw York, an at Ponc Lon Inlt, Floria, uing a boat-mount acoutic Dopplr currnt profilr. Dicharg pr unit with wa timat from th maurmnt by intgrating th vlocity profil ovr th pth. Profiling tranct acro th inlt throat occurr at or aroun th maximum bb or floo flow. Th rult ar hown in Figur 1 whr calculat icharg pr unit with i plott vru th trm 8 8 ([ g ( S 1) ] ) 1/ 3 / h 9 / on th right-han i of Equation 4. Grain iz for th Shinncock Inlt channl wa takn a 0.6 mm, whra a iz of 0.1 mm wa u for Ponc Lon Inlt. Both an wr aum to hav th am nity a quartz. Th ata point on Figur 1 how a wi rang of icharg pr unit with maur at th iffrnt pth. Howvr, thr i an uppr limit to th ata a inicat by th traight ah lin. Thi ah lin rprnt th maximum icharg pr unit with (q ) that can b utain at a particular valu of th paramtr 8 8 ([ g ( S 1) ] ) 1/ 3 / h 9 /. Th icharg inicat by th ah lin i trm th quilibrium maximum icharg. Any incra in icharg byon th quilibrium valu will rult in an incra in watr pth. Th cattr of maurmnt bnath th ah lin i pronounc, an thi inicat that th icharg calculat for tho maurmnt wa l than coul b tolrat by th pth at that location. Point jut bnath th ah lin might b location whr th prnt bottom wa ro by icharg lightly gratr than tho maur uring th fil xrci. Many of th ata point wll blow th lin cam from inlt cro ction ithr lightly awar of th jtti whr pth ar controll by wav an longhor currnt or lanwar of th ntranc channl whr th tial currnt i inufficint to cour th channl an pth hav bn incra by rging. 3

5 Figur 1. Fil ata from two ual-jtti inlt Anothr xplanation for ata cattr blow th ah lin i that pth at om of th location ar cour by a iffrnt cro-channl flow itribution that occur uring th rvr maximum tial flow. Finally, thr i th poibility that om of th pth ar th rult of couring that occurr uring pioic vnt uch a torm urg or rivr icharg combin with bb flow. Rgarl of th raon, pth aociat with ata point blow th ah lin ar not in quilibrium with th maur icharg. In othr wor, th pth woul b abl to accommoat incra flow icharg without aitional couring of th bottom. Th ah lin in Figur 1 corrpon to C = 5.1 in Equation 4, which can now b xpr a an mpirical quation for quilibrium maximum icharg pr unit with, i.., q 8 9 / 8 [ g ( S 1) ] 1/ 3/ = 5.1 h (6) 4

6 For a givn noncohiv imnt, thr i an quilibrium cour pth, h, aociat with th quilibrium icharg q. Th pth h i takn rlativ to th ti lvl at maximum icharg. An xprion for h i obtain by rarranging Equation 6 to gt h 0.34 q 8 / 9 = (7) 9 1/ 3 [ g ( S 1) ] 4 / Although it might b poibl to hav pth gratr than th quilibrium cour pth, th pth woul hav to b cau by om proc othr than th maximum icharg at that location. Etimat of quilibrium cour pth from Equation 7 houl b conir conrvativ bcau th timat rprnt th outr nvlop of th fil ata. In rality, th maximum icharg pr unit with may not prit long nough to allow cour pth to rach th prict quilibrium pth. Finally, ubtitution of th valu of C into Equation 3 an rarranging provi a rlationhip for man vlocity at a location in trm of th quilibrium pth an an paramtr, i.., V 8 1/ 8 [ g ( S 1) ] 1/ 3/ h = 5.1 (8) Plot of Equation 7 an 8 for a varity of quartz an iz ar givn in Figur an 3. Th plot how quilibrium pth (h ) a a function of quilibrium icharg pr unit with (q ) an man flow vlocity ( V ), rpctivly, for a rang of quartz an mian grain-iz iamtr. Th plot illutrat th ffct of grain-iz iamtr on th quilibrium pth. A xpct, channl with coarr an hav l pth at quilibrium unr th am flow conition. APPLICATION OF THE EQUILIBRIUM DISCHARGE DEPTH RELATIONSHIP: Th mimpirical rlationhip for quilibrium pth a a function of an paramtr an icharg pr unit with (Equation 7) or man vlocity (Equation 8) giv pth timat that ar probably conrvativ, i.., pr than might actually occur for th pcifi icharg. U of th formula houl b rtrict to rgion in th inlt throat whr th cour appar to b cau by th maximum icharg. For xampl, pth in cour hol form by vortic aociat with flow paration will not b prict by th quilibrium icharg pth rlationhip. In aition, th quation o not account for pth incra bcau of wav action in th channl. Important Not: Corrct u of th prictiv quation in thi CETN rquir that all variabl b givn in a conitnt t of unit. In particular, imnt grain iz n to b xpr in th am lngth unit u for q an g in th quation (mtr in th following xampl). 5

7 Figur. Equilibrium pth a a function of quilibrium icharg 6

8 Figur 3. Equilibrium pth a a function of maximum man vlocity 7

9 Exampl 1: Scour at Vntura Harbor, California. During torm conition, longhor currnt flowing through a narrow gap btwn th North Jtty an Dtach Brakwatr at Vntura Harbor cau a cour hol with maximum pth of 9.5 m blow man lowr low watr (mllw). Figur 4 how a plan viw of th navigation tructur. Th cour hol wa fill with quarryton an protct with a ton ill having top lvation at -4.5 m mllw (Hugh an Schwichtnbrg 1998). Th an in th vicinity i quartz with a mian grain iz nar = 0. mm (or = m). Figur 4. Vntura Harbor, California, navigation tructur An timat of th quilibrium icharg pr unit with corrponing to a cour pth of 9.5 m i trmin from Equation 6 uing a grain iz of 0. mm, i.., q = [( m / ) (.65 1) ] 1/ ( m) 3 / ( 9.5 m) 9 / = m / 8

10 Altrnatly, a valu for q can b ra irctly from th plot in Figur by fining th pth of 9.5 m on th vrtical axi, xtning a horizontal lin to intrct with th = 0. mm lin, an raing th corrponing icharg on th horizontal axi. Figur giv a valu of q 10.5m / Th maximum man vlocity corrponing to th quilibirum icharg i foun from Equation 5 a q 10.5m / V cour = = = 1.1m / h 9.5m Th am rult coul hav bn ra irctly from th curv in Figur 3 corrponing to = 0. mm. Alo not th timat aum maximum icharg occurring at mllw. Onc th cour hol wa fill in an capp at th -4.5-m mllw lvation, a imilar torm proucing th am icharg through th gap btwn th North Jtty an Dtach Brakwatr will prouc an incra man vlocity givn by V q h 10.5m / = 4.5m =.4m ill = / Viual timat of flow p through th gap uring a torm aftr th ill wa plac wr on th orr of m/. 1 Th timat of incra man vlocity i uful for trmining th abolut minimum cour blankt ton iz if cour hol ar fill an covr ovr with ton. Exampl : Frhwatr Dicharg. A rcnt moification to th jtty ytm of a fictitiou tial inlt on th Pacific Coat rult in bb-flow rirction an th formation of a 6-m-p cour hol ajacnt to on of th jtti. Th b matrial i quartz an with. 0.6 mm ( m). Th cour hol in it prnt configuration o not thratn th jtty to. During normal conition, only minor frhwatr runoff mpti into th bay an flow out th ntranc channl. Howvr, uring El Niño yar, a larg quantity of frhwatr runoff flow into th bay via floo channl. It i timat that th maximum frhwatr runoff will incra th icharg pr unit with through th inlt throat by 3 m / for a prio lating vral ay. What will b th maximum pth of th cour hol a a rult of th frhwatr urcharg? 1 Pronal Communication, 1998, B. R. Schwichtnbrg, U.S. Army Enginr Ditrict, Lo Angl, Lo Angl, CA. 9

11 Uing th plot in Figur, an quilibrium cour pth (rlativ to ti lvation of maximum icharg) of h = 6 m on th vrtical axi corrpon to a icharg pr unit with of q. 9.7 m / for an with mian iamtr of 0.6 mm = m. Aing th frhwatr icharg of 3 m / giv a nw icharg of 1.7 m /. From th am plot, thi incra icharg corrpon to a nw quilibrium cour pth of 7.7 m, or a 1.7-m pth incra. Thi timat i likly conrvativ bcau th combin frhwatr an bb-tial flow ar maximum for a rlativly hort tim uring ach tial cycl. Whthr or not vral ay will b ufficint tim to rach a nw cour quilibrium i unknown. Alo not that any ffct of flow tratification bcau of th influx of l n frh watr i not conir. Exampl 3: Nonquartz Simnt. Th curv in Figur an 3 prtain to quartz an with pcific gravity of.65. Etimat for inlt having nonquartz imnt mut u Equation 7 or 8. For xampl, conir anothr fictitiou tial inlt at a location whr th imnt i primarily brokn hll matrial having pcific gravity of S = ñ /ñ w =.4 an = 0.5 mm = m. What will b th quilibrium pth corrponing to a icharg pr unit with of 8 m /? Subtituting numrical valu for th variabl in Equation 7 yil 8 / ( 8m / ) 4 / 9 [( m / ) (.4 1) ] ( m) 0.34 ( 6.35) ( 3.04)( ) h = = m = 5. 8m An quivalnt timat for quartz an prouc an quilibrium cour pth about 0.4 m l than foun for th hll imnt. Thi iffrnc i probably l than th rror that might b xpct for thi primitiv timation tchniqu, o u of Figur an 3 for nonquartz imnt houl not introuc too much rror xcpt for xotic imnt that ar ithr xtrmly havy or narly nutrally buoyant. 1/ 3 ADDITIONAL INFORMATION. Qution about thi CETN can b ar to Dr. Stvn A. Hugh (Voic: , FAX: , -mail:.hugh@crc.w.army.mil). For information about th Coatal Inlt Rarch Program, pla contact th Program Managr, Mr. E. Clark McNair (Voic: , -mail: mcnairc@x1.w.army.mil). Dr. Nichola Krau provi a bnficial rviw of thi CETN. Thi tchnical not houl b cit a follow: Hugh, S. A. (1999). Equilibrium cour pth at tial inlt, Coatal Enginring Tchnical Not CETN IV-18, U.S. Army Enginr Rarch an Dvlopmnt Cntr, Vickburg, MS. 10

12 REFERENCES: Hugh, S. A., an Schwichtnbrg, B. R. (1998). Currnt-inuc cour along a brakwatr at Vntura Harbor, CA Exprimntal tuy, Coatal Enginring 34, 1-. Yalin, M. S. (1971). Thory of hyraulic mol. MacMillan Pr, Lonon. 11

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