NUMERICAL HYDRO- MORPHODINAMIC 2DH MODEL FOR THE SHALLOW WATERS

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2 Unverstà Poltecnca delle Marche Dpartmento d Idraulca Strade Ambente e Chmca Scuola d Dottorato d Rcerca n Scenze dell Ingegnera Currculum n Ingegnera de Materal delle Acque e de Terren NUMERICAL HYDRO- MORPHODINAMIC DH MODEL FOR THE SHALLOW WATERS Advsor: Prof. Ing. Alessandro Mancnell Ph.D. Dssertaton of: Matteo Postacchn Co-Advsor: Prof. Maurzo Brocchn Supervsor of the Ph.D. School: Prof. Grazano Cerr IX edton - ne seres

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4 Unverstà Poltecnca delle Marche Dpartmento d Idraulca Strade Ambente e Chmca Scuola d Dottorato d Rcerca n Scenze dell Ingegnera Currculum n Ingegnera de Materal delle Acque e de Terren NUMERICAL HYDRO- MORPHODINAMIC DH MODEL FOR THE SHALLOW WATERS Advsor: Prof. Ing. Alessandro Mancnell Ph.D. Dssertaton of: Matteo Postacchn Co-Advsor: Prof. Maurzo Brocchn Supervsor of the Ph.D. School: Prof. Grazano Cerr IX edton - ne seres

5 Unverstà Poltecnca delle Marche Dpartmento d Idraulca Strade Ambente e Chmca Va Brecce Banche Ancona Italy

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8 Acknoledgements I sh to thank all the research team of the Dpartmento d Idraulca Strade Ambente e Chmca Sezone Idraulca for the support durng the entre three-years perod of PhD. In partcular I ould lke to thank Prof.Ing. Mancnell and Prof. Brocchn for the essental contrbuton n the development of ths ork for both numercal and expermental part. Then I ould lke to thank for ther precous and helpful ad Marc Landon and Camlle Chauvgné ho contrbuted to the development of the numercal hydro-morphodynamc solver and Dott.Ing. Lorenzon ho orked th me at the experments performed n the ave flume of the Hydraulc Laboratory of the Unverstà Poltecnca delle Marche of Ancona. Fnally thank to Prof. Fraccarollo for the useful data provded for the valdaton tests of the numercal solver.

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10 Abstract In the present thess a hydro-morphodynamc numercal model s llustrated as a novel contrbuton to the nvestgaton and predcton of nearshore flos and seabed changes forced by aves and currents. The model ncludes a robust hydrodynamc solver for the ntegraton of the Nonlnear Shallo Water Equatons (NSWE) and a rather flexble solver for the resoluton of the Exner equaton (used to evaluate the morphologcal evoluton of the sea bottom). Couplng of NSWE and Exner equaton and updatng of the soluton s made by means of a sequental splttng scheme. The model has been valdated by reproducng both numercal/analytcal tests present n the lterature of the last years and laboratory experences performed n the Hydraulc Laboratory of the Unverstà Poltecnca delle Marche (AN). The smulaton of the exstent theoretcal solutons led to consstent results for hat concerns both hydrodynamcs and morphodynamcs especally n the predcton of the seabed evoluton due to ether bed-load or suspended-load transport after dam-break and sash events. The comparson beteen numercal results of the solver and expermental data s partally exhaustve. In fact the solver reproduces farly ell the man bottom features n the presence of spectral aves but fals hen regular aves are forced because of laboratory effects occurred n the ave flume.

11 Contents Contents... Lst of Fgure s... Lst of Tables... v Chapter.... Introducton..... Wave dynamcs n the nearshore zone..... The modelng of the nearshore sedment transport The hydrodynamcs equatons The Exner equaton The couplng ssue A fully-coupled analytcal/numercal model of bed-load transport: velocty-based la (Kelly & Dodd 00) A fully-coupled analytcal/numercal model of bed-load transport: velocty- and depth-based la (Kelly 009) An analytcal decoupled model of suspended sedment transport (Prtchard & Hogg 005) A numercal fully-coupled model of sheet-flo transport (Fraccarollo & Capart 00)... 0 Chapter The numercal solver Splttng of the system and couplng ssues The hydrodynamc solver System resoluton Integraton: fnte-volume dscretzaton The WAF method Shorelne treatment The morphodynamc solver Sedment transport formulatons System resoluton... 7

12 .3.3. Flux computaton Shorelne treatment Numercal nstabltes and flterng A lo-pass flter: Shapro flter A second lo-pass flter: the targeted flter... 9 Chapter The expermental tests The expermental set up Salent morphologcal results Features of the seabed profles Seabed evoluton Chapter Valdaton of the numercal solver Valdaton of the bed-load transport: Fraccarollo & Capart (00) Numercal results from the hydro-morphodynamc (HM) solver Valdaton of the bed-load transport: Kelly (009) Valdaton of the suspended-load transport: Prtchard & Hogg (005) Valdaton of the total transport: flume experments Numercal results for structure-free confguraton (G) Numercal results for the submerged-breakater confguraton (B) Comparsons th expermental results (confguraton G) Comparsons th expermental results (confguraton B)... 6 Chapter Concludng Remarks References Appendx A The theoretcal approach n the laboratory experments... 7 Appendx B Rgd bodes n the doman... 73

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15 Lst of Fgures Fgure Sequental computaton n CAM models (adapted from Dng et al. 006). Fgure Vertcal flo structure; from the top to the bottom: pure-ater layer flong mxture of ater and grans granular substratum. Γ Γ s and Γ b are the nterfaces beteen ar and ater ater and mxture mxture and bed respectvely (adapted from Fraccarollo & Capart 00). Fgure 3 Sketch of seabed profle and free surface elevaton. Fgure 4 Flo chart of the computatonal schemes. Fgure 5 Sketch of the doman S n the x t space. Fgure 6 Flux computaton from one specfc Remann problem n the x t space. Fgure 7 Sketch of the soluton of the shorelne Remann problem and analogy th the movng pston. Fgure 8 Bathymetrc map (left) and reproduced movable-bed model n the flume (rght). Fgure 9 Sketch of the flume: shorelne (left) breakaters at the tested postons (mddle) and ave generator (rght); S5 S6 and S7 gve the postons of three ave gauges. Fgure 0 Chronologcal evoluton of the sea-storm OS th ave characterstcs (Hs Tp drecton and duraton) of each phase at both prototype and model condtons. Fgure Superposton of the bed profles after each phase of the sea storm OS3 (September 004) for confguraton D. Fgure. Comparson among the fnal profles due to OS for emerged confguratons A and D (green and magenta lne top panel) and for submerged confguratons B and C (red and yello lne bottom panel); the black dotted lne represents the ntal profle; horzontal and vertcal scales are n mllmeters. Fgure 3 Snapshots from the expermental tests of Tape at t = 0s 0.s 0.s 0.3s 0.4s 0.5s (left panels from the top to the bottom) and Louvan at t = 0.5s 0.50s 0.75s.00s (rght panels from the top to the bottom adapted from FC0). Fgure 4 Results from Tape tests after t = 0.5s : black lnes gve the expermental results green lnes the results by FC0 red lnes numercal solutons of the HM solver; sold lnes refer to the stll ater level Γ dash-dotted lnes to the Γ s lmt dashed lnes to the bottom level Γ b. Fgure 5 Results from Louvan tests after t = s : black lnes gve the expermental results green lnes the results by FC0 red lnes numercal solutons of the HM solver; sold lnes

16 refer to the stll ater level Γ dash-dotted lnes to the Γ s lmt dashed lnes to the bottom level Γ b. Fgure 6 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Au3 here A = 0 3 sm. Fgure 7 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Au3 here A = sm. Fgure 8 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Au3 here A = 0 sm. Fgure 9 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Adu3 here A =.5 0 sm. Fgure 0 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Adu3 here A = sm. Fgure PH05 soluton for the net fluxes Q ~ (panel a) and the nstantaneous flux q ~ (panel b): (a) dashed lnes gve the net fluxes occurrng durng the uprush (postve values) and the backash (negatve values) the sold lne gves the net flux over a cycle; (b) sold lnes gve the flux at samplng ponts located at x ~ = (adapted from PH05). Fgure Spatal evoluton of Q ~ (panels a c e) and temporal evoluton of q ~ (panels b d f) from the HM solver obtaned by usng a frcton coeffcent C τ = 0.00 (top panels) C τ = (mddle panels) and C τ = 0.0 (bottom panels). Fgure 3 HM-solver results: cross-secton seabed evoluton n the absence of structures (confguraton G) under the rregular ave forcng OS. Fgure 4 HM-solver results: cross-secton seabed evoluton n the absence of structures (confguraton G) under the regular ave forcng OR. Fgure 5 HM-solver results: cross-secton seabed evoluton n the presence of a submerged structure (confguraton B) under the rregular ave forcng OS. Fgure 6 HM-solver results: cross-secton seabed evoluton n the presence of a submerged structure (confguraton B) under the regular ave forcng OR. Fgure 7 Seabed evoluton for confguraton G under the regular ave forcng OS: comparson of the fnal results of the HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. Fgure 8 Seabed evoluton for confguraton G under the regular ave forcng OR: comparson of the fnal results of the HM solver (sold red lne) and flume experments v

17 (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. Fgure 9 Seabed evoluton for confguraton B under the rregular ave forcng OS: comparson of fnal results comng from HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. Fgure 30 Seabed evoluton for confguraton B under the regular ave forcng OR: comparson of fnal results comng from HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. Fgure 3 Sketch of the flos nterestng the plng-up phenomenon n the presence of emerged (top fgure) or submerged (bottom fgure) breakaters. Fgure 3 Representatve sketch of the three-layer dstrbuton: sea level movng seabed and rgd seabed. Fgure 33 Sketch of fluxes enterng and escapng the cell ( ) at tme n/. v

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19 Lst of Tables Table Man geometrcal characterstcs of the tested breakaters at model scale. Table Characterstcs of the reproduced aves at model scale. v

20 Chapter. Introducton.. Wave dynamcs n the nearshore zone The marne regon that s the closest to the emerged beach s called nearshore zone. Its dynamcs s domnated by many parameters that are nvolved n several physcal processes concernng both hydrodynamcs and morphodynamcs. The man forcng of all these processes are aves and the nduced currents that generate sgnfcant velocty felds actng drectly on the sea bottom and provdng mportant morphologcal changes. In partcular surface gravty aves propagatng across the contnental shelf to the beach ave-nduced currents n the surf zone longshore and cross-shore sedment motons are the phenomena that nduce most of the beach evoluton. Knoledge of all these processes s fundamental for beach eroson predcton nourshment management mantenance and dredgng of navgaton channels and desgnng of shore protecton structures. The forecast of the beach evoluton for hat concerns accreton/evoluton processes s noadays one of the man topcs and targets of coastal research. As descrbed by Hamm et al. (993) the ave propagaton from the offshore to the coast nduces large hydrodynamc and morphodynamc changes especally hen the ater depth s not large and nterferences th the seabed become mportant. The seabed varaton nduces effects such as refracton dffracton shoalng and breakng. The latter s such a strong phenomenon that leads to ave reducton and seabed evoluton n vrtue of the energy transfers nduced by such a dsspatve phenomenon. In that porton of the nearshore regon that s commonly called surf zone many processes connected th breakng take place: frst an ntal decay of the aves and a rapd change of ther shape occurs because of ave breakng (n the outer surf zone ); then a slo evoluton of the aves and the achevement of a quas-steady state smlar to that of movng hydraulc umps (thn the nner surf zone ); fnally run-up and run-don events n correspondence of the sash zone. All these processes forced by the aves approachng the shore nduce strong ncreases of the velocty feld hch n turn force sgnfcant sedment motons especally n the sash zone here the ave uprush and backash lead to a relevant seabed evoluton. At

21 the same tme bottom changes nduce sgnfcant varatons of the hydrodynamc condtons. In other ords a feedback mechansm exsts such that the hydrodynamcs affect the morphodynamcs and vce versa... The modelng of the nearshore sedment transport Durng the last decades many progresses have been made n the study of the nearshore morphodynamcs because several experences have been carred out all over the orld both n the laboratores and drectly n the feld. Also many numercal and analytcal models have been developed and upgraded n the years on the bass of the physcal and mathematcal tools that are avalable for the correct representaton of aves currents and sedment transport. The am to gve the best representaton of the beach evoluton nduced by the hydrodynamc feld due to both aves and currents occurrng n the nearshore zone has been pursued th more ncsveness n the last decades and numercal smulatons are becomng an ndspensable tool to reach such a scope. As an example ave breakng s a very complex phenomenon but t can be reproduced by means of hydrodynamc solvers that enable to smulate the breaker propagaton through the surf zone and ts evoluton from the breakng lne to the shore. Hoever the natural morphologcal processes and the mechansms of sedment transport have not been fully understood nether descrbed adequately by physcal prncples and mathematcal analyses (Dng et al. 006). In order to smulate the morphologcal processes thn reasonably short computatonal tmes a seres of constrants and approxmatons can be appled to the numercal solver. De Vrend et al. (993) classfed the numercal models for practcal smulatons of morphologcal processes nto four types: ) one-dmensonal (D) longshore coastlne models; ) to-dmensonal (DV) cross-shore coastal profle models; 3) to-dmensonalhorzontal (DH) morphologcal models; 4) fully three-dmensonal (3D) local morphologcal models. Solvers of the frst group enable the users to study the evoluton of both longshore sedment transport and shorelne. DV cross-shore models predct only the vertcal varatons of coastal profles but not the varatons of the longshore sedment transport. DH models enable to reproduce all the phenomena occurrng n the nearshore area by means of an average of the vertcal dstrbutons of all the varables nvolved n both ave and current processes. The numercal solvers of class 4) of above are the most complete and together th the DH models they represent all the longshore and cross-shore morphologcal changes but also take nto account the vertcal dstrbutons of all the terms that are nvolved n these processes (.e. velocty sedment transport etc). Hoever fully-3d morphologcal models are usually used to predct the hydromorphodynamc evoluton of a relatvely small feld and over a short perod. In fact

22 because of ther complexty the computaton tme s large and they appear not suted to reproduce a long event actng n a large spatal doman. In the recent perod the evoluton of morphodynamc solvers focused especally on the predcton of to-dmensonal horzontal patterns that are related to the velocty feld nduced by ater motons. Hence DH depth-averaged models can ell represent bed evoluton n large-scale areas even f also the quas-3d models found a good success n the last decades. As an example Zyserman & Johnson (00) orked on a quas-3d model here an emprcal 3D shear stress dstrbuton allos to take nto account the threedmensonal effect of sedment transport hle Dng et al. (006) developed a quas-3d model (Q3DCAM) that s both robust as a DH model and able to nclude the effects of the vertcal varatons of currents due to the surface roller characterzng breakng aves. For hat concerns the classcal DH depth-averaged models for the nearshore zone De Vrend (987) clamed that they are typcally characterzed by to man modules each provdng a dfferent computaton n to separated steps. Durng the fxed-bottom step ater and sedment motons are evaluated thout provdng any modfcaton of the bed that s taken as rgd. The changng-bottom step conssts n the evaluaton of the bottom varaton hle the other varables are fxed. As a result the strong approxmaton due to the separated treatment of hydrodynamcs and morphodynamcs does not nduce a perfect representaton of nature but at the same tme computatonal tmes are small. In summary t seems that hat the approach chosen gves s the best compromse beteen approprate numercal solutons and reduced computatonal costs. As a consequence of ths need dfferent types of numercal solvers based on separated modules rose n the last decades th the am to study separately all processes (aves current sedment transport) and to update them n a compound morphologcal model (De Vrend et al. 993). The connectons among all modules and the ay to represent the processes leads to more or less complcated codes ( Intal Sedmentaton/Eroson models Medum-Term Morphodynamc models etc). As an example of the numercal structure presented above Coastal Area Morphologcal (CAM) models can be consdered. In order to smulate at best the morphodynamc changes and the shorelne evolutons (Dng et al. 006) a process-based approach s used n such models.e. ave feld current feld and seabed changes are computed sequentally and not at the same tme. The result s a ne bathymetry representng the bottom nput for the next tme step as shon n Fgure. Ths procedure s the bass of a ave-current-morphologcal model and enables to smulate the long-term morphologcal processes by usng an emprcal sedment transport model even f calbraton/valdaton of such a model s requred before applyng t to a real-lfe coastal problem. In fact t s necessary to check f the modelng of aves currents and sedment transport evaluates correctly the man processes occurrng durng the ave propagaton and f each module of the solver predcts th reasonable accuracy all the parameters of nterest. Furthermore t s mportant to have a code robust enough to smulate the long-term morphodynamc evoluton by selectng a reasonably long feedback perod T sed (see Fgure ). 3

23 Fgure Sequental computaton n CAM models (adapted from Dng et al. 006).... The hydrodynamcs equatons In the attempt to reproduce at best the hole hydrodynamcs to man approxmatons of the classcal Naver-Stokes equatons for the nearshore zone are used: the Nonlnear Shallo Water Equatons (NSWE) and the Boussnesq-type equatons. In the last decades the fashonable Boussnesq-type models led to very nterestng solutons n several coastal applcatons though such a modelng as stll lmted because of some mportant problems. In fact the treatment of phenomena lke the shorelne moton the ave breakng and the propagaton of breakng fronts ere not adequate. Hence many studes ere dedcated to mprove the mentoned lmtatons and a more mature generaton of Boussnesq-type models s currently avalable hch better represents breakng aves (e.g. see Brocchn et al. 99) and frequency dsperson propertes (e.g. see Madsen & Schäffer 998) to extend the models valdty to the surf zone and to the deep aters respectvely. NSWE models are very good for smulatng both ave propagaton over the nner surf zone and sash zone. Wthn such models ave breakng s represented as a 4

24 dscontnuty of the ater flo. Ths approach gves at the same tme a smple nterpretaton of a very complcated phenomenon and an ncorrect locaton of the breakng pont hch s alays placed too far offshore than hat found n experments. Such a dsadvantage s due to the absence of frequency dsperson so that the predcted hydrodynamcs n the outer surf zone s not completely consstent th nature. Hoever applcatons of NSWE solvers to nearshore flos often lead to good results because of the quas-steady state of the shallo-ater breakers.... The Exner equaton The morphodynamcs can be farly ell descrbed by means of the Exner equaton that represents the sold mass conservaton la. Startng from the velocty feld such a conservaton la enables to predct the seabed changes by means of the sedment transport varaton over the horzontal plane. Usually the nearshore sedment can be moblzed n three dfferent modes (as descrbed n Fredsøe 993): bed-load sheet flo and suspendedload. They dstnctly characterze dfferent portons of the ater column that s nterested by the sedment moton. The frst mode of transport refers to sedments that are n almost contnuous contact th the bed and depends drectly on the bed shear stress that should be small enough for ths mode to occur. Suspended sedments move through the hole ater column and are forced by the ater flos. The sheet flo s a specfc transport mode that s can be descrbed as beng n beteen suspenson and bed-load transport. It concerns sedments movng over the bed thout any contact n a ell-defned layer and occurs hen shear stresses are rather large. Generally numercal models represent the sedment transport as the sum of only to contrbutons: bed-load and suspended-load transport thout takng nto account the sheet flo moton. Many formulas have been developed n the recent years and dscussons about them (e.g. Camenen & Larson 008) suggest the applcaton feld for each relaton and also hch ones are most suted to be mplemented n a model. In order to obtan both hydrodynamc and morphodynamc evolutons a eakly-coupled model bult on a fnte-volume scheme has been developed based on varous sedment transport formulatons for both bed-load and suspended-load..3. The couplng ssue A large amount of studes concernng the morphodynamcs n both surf and sash zone have led to the development of several mathematcal models n the last years. They are characterzed by dfferent features and dfferent complextes dependng on the am of the solver at hand. Several studes have been carred out to understand the behavor of the sash zone dynamcs under ether perodc or mpulsve (dam-break type) ave forcng. In fact because many varables are nvolved n the large amount of processes concernng the 5

25 sash moton (such as sedment transport groundater dynamcs beachface morphology ater-table contrbuton etc for more detals see Masselnk & Puleo 006) the sash zone represents one of the most dffcult areas to be descrbed of the entre nearshore. Because of the nteracton among all the nvolved parameters the correct reproducton of the sash zone morphologcal evoluton s rather dffcult and many models both numercal and analytcal have been bult to smulate such a dynamcs as ell as possble by applyng sedment transport formulas dependng on one or both shallo-ater hydrodynamc varables (depth and depth-averaged velocty). In the attempt to provde a farly good llustraton of the state-of-the-art of mathematcal modelng concernng the nearshore zone n the follong subsectons a specal attenton s gven to the descrpton of some of the most recent orks that are avalable n the lterature. Decoupled and coupled as ell as analytcal and numercal models are analyzed. It s mportant to note that the equatons used to descrbe the hydrodynamc processes occurrng n the nearshore area.e. the NSWE are shared by all the consdered models..3.. A fully-coupled analytcal/numercal model of bed-load transport: velocty-based la (Kelly & Dodd 00) The ork of Kelly & Dodd (00) s based on the descrpton of a mathematcal model n hch hydrodynamcs and morphodynamcs are drectly coupled. The result s that seabed and flo evoluton are ntegrated thn the same tme step and thn the constrants mposed by the nvscd shallo-ater dynamcs and by a smple sedment transport relaton. An uncoupled model has also been developed n order to understand the dfferences exstng th the fully-coupled one. Furthermore the model does not take nto account contrbutons of ater nfltraton/exfltraton sedment storage settlng lag advecton of sedment from the surf zone and bed shear stress hch are fundamental processes occurrng on the beachface durng sash events (e.g. see Brocchn & Baldock 008). The mathematcal model predcts the seabed changes and the velocty felds as forced by a sngle sash event and enables a comparson th the analytcal soluton of Shen & Meyer (963). The flo feld n the sash zone s ell represented by the NSWE that n dmensonal form are rtten as: d d u u d = 0 (.) t x x ( d z) u u u g t x x = 0 (.) here d(x t) s the total ater depth u(x t) the depth-averaged flo velocty along the horzontal x-drecton z(x t) the beach surface level and g s gravty acceleraton. 6

26 The sedment transport equlbrum s gven by the contnuty equaton for the sold mass.e. by the follong formulaton of the Exner equaton: z q ξ = 0 (.3) t x here q s the horzontal sedment flux and ξ = ( p) a coeffcent dependent on the beach porosty p. Generally the sedment transport contrbuton depends on both velocty and nstantaneous ater depth.e. q(u h) but n the present model a smplfed sedment flux dependng on u only s used: 3 q = Au (.4) here A s a dmensonal constant that can be taken as a functon of sedment characterstcs (medan dameter d 50 relatve densty s) and frcton coeffcent f R as suggested by Kelly (009) n the follong formula: f R 3 A 8 = g( s ). (.5) The choce of such a formula s based on ts smplcty and on the consequent possble applcaton n sem-analytcal modelng. By substtutng the closure equaton (.4) nto (.3) the follong result s found: z u 3Aξ u = 0. (.6) t x In the uncoupled approach the hydrodynamcs s ntegrated separately from the morphologcal evoluton so that the beachface s updated at the end of each sash event by consderng the net sedment flux gradent at each pont. In order to reproduce Shen & Meyer (963) s test equaton (.6) s ntegrated at each pont over the beach beteen the nundaton tme (t n ) and the denudaton tme (t de ) so that the net sedment flux Q s obtaned over an entre sash cycle. Because of the uncouplng the seabed s not updated durng the sash event and n the case of a plane beach characterzed by a slope β the term z x = tan β remans constant n (.). For the fully-coupled model the NSWE and the Exner equaton are solved smultaneously therefore the evoluton of the beachface has an mmedate mpact on the hydrodynamcs and vce versa. By means of standard soluton technques (Stoker 957) the NSWE (.) and (.) are combned together and solved for a typcal Remann problem n terms of characterstc varables for an nvscd hydrodynamc flo as better explaned n..3. Instead n order to obtan the characterstc equatons for the hole shallo-ater-exner system.e. the couplng of both hydrodynamc and morphodynamc evolutons 7

27 combnaton of the governng equatons (.) (.) and (.3) leads to dervatves n a sngle drecton of the three nvolved varables as llustrated n Kelly & Dodd (00). It can be expressed by solvng for the characterstc drectons λ k : 3 k k λ uλ [ u g( d 3Aξu )] λ 3Aξgu = 0 for k = 3 (.7) The Remann equatons along these characterstcs are: k 3 du g dd g dz = 0 for k = 3. (.8) dt ( λ k u) dt λ k dt For physcally-realstc stuatons.e. th h 0 the dscrmnant of (.7) s less than zero hence the governng equatons are hyperbolc th real ave speeds (represented by the egenvalues λ k )..3.. A fully-coupled analytcal/numercal model of bed-load transport: velocty- and depth-based la (Kelly 009) The present model s bult on the same hydro-morphodynamc system of equatons presented n.3. except for the closure equaton that s used to descrbe the sedment transport. Hence equatons (.) (.) and (.3) reman the same hereas (.4) changes nto: 3 q = Au d (.9) here A s stll a dmensonal constant lke the parameter A of the prevous model (see equaton (.4)) that can be consdered as a functon of the sedment characterstcs and bedshear stress. Use of the present formula s a bt more complcated than for (.4) though stll useful a sem-analytcal treatment. Substtuton of (.9) nto (.3) leads to: z 3 d u Aξ u 3Aξu d = 0. (.0) t x x The combnaton of the governng equatons (.) (.) and (.0) and the applcaton of the total dervatve leads to the characterstc polynomal: 3 k k λ uλ ( u gd 3Aξgu d) λ Aξgu d = 0 for k = 3 (.) and to the Remann equatons: k 3 8

28 du λ 3 k u Aξgu dd g dz = 0 for k = 3. dt (.) d λk dt λk dt Even n ths case for physcally-realstc stuatons (h 0) the dscrmnant of (.) s less than zero. The governng equatons are therefore hyperbolc and the ave speeds are real An analytcal decoupled model of suspended sedment transport (Prtchard & Hogg 005) In general the asymmetry of the flo durng a sash moton makes many sash models that use poer-la-based sedment transport formulas and th decoupled hydromorphodynamcs predct net offshore transport of the sedment everyhere on the beach. On the contrary feld studes ndcate that under certan condtons sngle sash events can produce a net onshore movement of sedments. In order to understand such a dfference n the sedment transport predcton Prtchard & Hogg (005) developed an uncoupled analytcal model that ncludes terms for presuspended sedment and settlng lag n order to take nto account the contrbuton from sedment entraned thn the sash zone and that one from the sedment suspended by the ntal bore collapse. The ncluson of these terms led to the possblty of local net onshore transport of sedment. As n the prevous models smulaton of the flud moton s acheved by means of the NSWE. In ths case mass and momentum equatons are rtten n terms of the ater depth d that s perpendcular to the bed and the depth-averaged velocty u that s parallel to the bed. Further the sedment transport s descrbed by the only suspended-load contrbuton that depends on a depth-averaged mass concentraton c. The horzontal coordnate x s parallel to the bottom even f nclned. The set of equatons to be solved s thus rtten as follos: ( ud ) d t x = 0 (.3) u u d u g cos β = g sn β (.4) t x x c c meqe sc u = (.5) t x d here m e q e s a mass eroson rate per unt area and s the effectve settlng velocty of sedment partcles. Equatons (.3) and (.4) represent the conservaton of flud mass and flud momentum respectvely hle (.5) s a depth-ntegrated transport equaton for the suspended sedment and represents the sold mass conservaton. The horzontal dffuson of suspended sedment s neglected because of ts small contrbuton n shallo aters th 9

29 respect to the advectve transport as also percolaton of ater nto and out of the beachface s neglected. The rght-hand sde of (.5) represents the eroson/deposton of sedment here the mass eroson rate q e s a functon of the hydrodynamc condtons and s evaluated as: m q here n > 0 s a parameter and e e = m e τ τ e τ 0 n (.6) τ = c D ρ u u the bed shear stress; τ e and τ 0 are respectvely a threshold stress for eroson and a reference shear stress: f τ < τ e no eroson occurs. The nstantaneous and net sedment fluxes are defned as: q ( x t) = h( x t) u( x t) c( x t) (.7) t t de Q( x) = q( x t) dt (.8) n here t n and t de represent respectvely the nundaton and denudaton tme durng a sash event so that Q(0) s the net mass of sedment per unt cross-stream dth hch s mported to the sash zone by ths event. The decoupled analytcal model ell descrbes the sash motons and the velocty feld nduced by a bore-collapse on a planar nclned beach. The authors valdated the model by comparng ther results th the hydrodynamc exact soluton by Shen & Meyer (963) obtanng a good agreement. Concernng the sedment transport behavor durng a sash event Prtchard & Hogg s model enables to smulate both steady condtons.e. thout any transport addton and non-steady condtons.e. th the possblty to account for pre-suspended sedments or lag effects of sedment transport. In Prtchard & Hogg (005) the response of sedment transport to dfferent drag coeffcents c D s also nvestgated A numercal fully-coupled model of sheet-flo transport (Fraccarollo & Capart 00) The sudden erosonal flos that are essentally due to dam-break events can be descrbed by means of an approprate set of equatons and consttutve las. It s thus mportant to retan mportant features such as the thckness of the sedment transport layer and ts nerta lke n classcal alluval hydraulc treatments. Wth ths perspectve Fraccarollo & Capart (00) studed the flo n the vertcal plane by approxmatng t as made of regons of homogenous propertes separated by thn nterfaces.e. applyng a dscrete system approach. In partcular the nterface representng the bed boundary snce t s a phase nterface across hch the saturated sedment materal undergoes a transton from sold- to flud-lke behavor t s characterzed by mportant features and s accurately descrbed n the model governng equatons. 0

30 The model of Fraccarollo & Capart (00) s focused on regmes of ntense eroson and sedment transport larger than Shelds' threshold for gran moton. Wthn such regmes coarse sedments move collectvely as a sheet of contact load occupyng a sgnfcant porton of the flo depth. Such a transport layer can ether be entraned by an upper layer made of clear ater or extend across the entre flo depth as n typcal debrs flos. Because of these consderatons the suspended-load s neglected n the model. In order to represent as correctly as possble a dam-break event four dfferent features characterze the vertcal dstrbuton of the flo regon: a shallo-ater layer an effectve mxture flo a contact load and a morphodynamc nterface. Referrng to Fgure three sharp nterfaces can be ntroduced. Γ s the ar-ater boundary at the flo free surface. Γ s defnes the upper lmt of the transport layer separatng the clear ater layer above from the lqud-granular mxture belo. Mature debrs flo condtons are obtaned hen nterfaces Γ and Γ s concde. The thrd nterface Γ b s the bed boundary that s totally dfferent n nature from the other to. Whereas Γ and Γ s are materal nterfaces that are passvely convected th the flo the actve boundary Γ b s modeled as a phase nterface across hch the mxture undergoes a dscontnuous change of state. Above Γ b the mxture flos as a flud hle a rgd granular skeleton s consdered underneath. Eroson occurs hen Γ b progresses donards and deposton results hen the t moves up. In both cases no vertcal moton of materal occurs but lquefacton or soldfcaton takes place provdng the bed nterface dsplacement. Fgure Vertcal flo structure; from the top to the bottom: pure-ater layer flong mxture of ater and grans granular substratum. Γ Γ s and Γ b are the nterfaces beteen ar and ater ater and mxture mxture and bed respectvely (adapted from Fraccarollo & Capart 00). Applng the Reynolds transport theorem (e.g. see Whte 999) to the balance of mass and momentum n a specfc control volume the follong ntegral equatons can be obtaned:

31 t x x t t x x x x h dx m h dx s [ h ( u v) ] = e dγ m m Γ b [ h ( u v) ] = e dγ s m Γ b [ gh rgh ] ( h rh ) u dx ( h rh ) u ( u v) m s m m s m m b b b b m = ρ Γb s bx = dγ b (.9) (.0) (.) t x x z dx b [ vz ] = e dγ b Γ b b b (.) here: x and z b are respectvely the horzontal and vertcal coordnates; x and x are the lateral postons of the control volume boundares; Γ b s the porton of Γ b n beteen the control volume; h m and h s are respectvely the total depth over the bed boundary and the thckness of the mxture layer (see Fgure ); u m s the horzontal mxture velocty; v s the non-materal boundary velocty; g and ρ are respectvely the gravty acceleraton and the ater densty; r = (s )φ s s the densty supplement due to the presence of the sedment load; e b s the eroson rate.e. the volume flux densty across the bed nterface Γ b ; bx s the x-component of the momentum flux densty across Γ b ; f obects n squared parentheses gve a ump across a dscontnuty: [ f ] f =. The above ntegral equatons represent the flo layer contnuty (.9) the transport layer contnuty (.0) the horzontal flo momentum (.) and the conservaton of mass appled to the sold bed substrate (.). Assumng that shocks are absent the flo s gradually vared and Γ b s gently slopng t s possble to rte the governng (smplfed) equatons n dvergence form by takng the lmt x x :

32 h t m t x s b ( h u ) = e ( h u ) = e = e m m h t x (( h rh ) u ) ( h rh ) m s m g z t b s m b b x here the eroson rate e b s gven by: e b = ( u gh rgh ) b bn ( h rh ) = ρ m m s ( r) s z x u m m τ ρ ( τ τ ) m bn m s (.3abc) (.4) (.5) hle τ bn and τ m are the shear stresses actng on both sdes of the bed nterface Γ b. The system of dvergence equatons (.3abc) and (.4) can be rtten n vector form as: U U A( U) = S( U) (.6) t x here U s the vector of dependent varables A(U) the Jacoban matrx and S(U) the source term vector. By means of splttng procedures the system can be decomposed nto three dstnct components each assocated th a specfc flo process. The flo hydrodynamcs s descrbed by the homogeneous PDE system hle geomorphc exchange and frctonal momentum loss processes are represented th the to ODE systems: U U A( U) = 0 t x (.7) U U = Ge ( U) = Fe ( U) (.8ab) t t here the source vector S(U) has been splt nto to dstnct components G e (U) and F e (U) and the to equatons assocated th them represent respectvely the geomorphc bed change drven by stress dfference beteen τ bn and τ m and the frctonal momentum loss due to resdual shear stress τ bn. Ne approxmatons such as τ bn = τ m = 0 enable to arrange the equatons reducng the above set as follos: 3

33 = = x u h t z x u h x h u t h m s s m m (.9ab) ( ) ( ) ( ) ( ) ( ) = x u u h r h h r h x z g h r h h r h x h g h r h h h t u m m s s s s s s s m (.30) here h = z z s s the depth of the pure-ater layer (see Fgure ) z s = z b h s and g u h m s / µ = the elevaton of the top of the sedment transport layer and ts thckness (beng µ a sedment moblty constant). Equatons (.9ab) represent respectvely a contnuty equaton for the pure- ater layer and an Exner equaton expressng conservaton of the total sedment volume hle (.30) s an equaton of moton for the heterogeneous flong mxture. The vectoral form of such a system s: 0 ) ( = x t W W B W (.3) that forms a set of quas-lnear frst-order partal dfferental equatons. The system soluton moves through dentfcaton of the egenvalues comng from matrx B. The fnal soluton can be found by solvng the problem at hand as a typcal Remann problem.

34 Chapter. The numercal solver As already mentoned the NSWE are depth-averaged equatons of conservaton of mass and momentum hch can be solved even hen hydraulc umps appear (e.g. see Whtham 974). A complete verson of the NSWE n a frst nstance ncludes seabed frcton. Indeed the effects of seabed frcton on nearshore dynamcs are partcularly sgnfcant both for near-shorelne flos and for longshore motons occurrng n the surf zone but ther representaton stll poses many theoretcal and practcal problems. Formulatons of the effects of bottom frcton as a bulk frctonal force usng sem-emprcal formulae lke the Chezy la have generally been successful even for the oscllatory motons typcal of the surf and sash zone (e.g. Longuet-Hggns 970; Watson et al. 994). The Exner equaton models sedment conservaton over the ater column. The evaluaton of the sedment fluxes there appearng s probably the most dffcult aspect to be modeled. Ths dffculty also derves from the huge number of closure las avalable to descrbe the sedment transport (e.g. Camenen & Larson 008). One common feature of almost all these las s to dstngush the near-bed contrbuton to the sedment transport (bed-load) from contrbutons accountng for the transport of sedments spread outsde the bottom boundary layer all over the ater column (suspended-load). The NSWE/Exner system rtten n conservatve form s a quas-lnear hyperbolc set of equatons. The standard method for solvng such a system s knon as the method of characterstcs. The Remann structure of the fully-coupled system requres heavy computatons for the egenvalues and Remann varables. The novel solver here llustrated s based on a rather dfferent perspectve. It s supposed that the uncertantes related th the choce/use of approprate sedment transport equatons s so much hgher than the eventual naccuraces assocated th the use of a eakly-coupled numercal solver that e have preferred to buld a theoretcal/numercal frameork hch enables a free choce of sedment transport las and does not: ) rely on a specfc sedment transport formula ) need to compute a specfc characterstc ave structure. Explct rtng and soluton of the approprate characterstc ave structure s only possble hen very smple sedment transport las are used (e.g. Kelly & Dodd 00). Theoretcally the draback of the approach here proposed s a eak couplng beteen hydrodynamcs (NSWE) and morphodynamcs (Exner). Hoever t s shon that 5

35 dfferently from hat conectured by some researchers (see Kelly 009 and Kelly & Dodd 00) a eakly-coupled model stll captures all the man features of nterest... Splttng of the system and couplng ssues The fully-coupled system n ts non-conservatve form s: d ( ud) ( vd) = 0 (.) t x y v u ut uux vu y gd x = gz x Cτ (.) d v v vt uvx vvy gd y = gz y Cτ (.3) d z λ ( P Q ) = 0 t x y (.4) here: (xyz) are Cartesan orthogonal coordnates beng the stll ater level z=0; d(xyt)=η(xyt) z(xyt) s the ater depth th η(x y t) and z(x y t) beng respectvely the free surface and the seabed poston th respect to the stll ater level; v=(uv) s the depth-averaged velocty vector (u(xyt) and v(xyt) beng respectvely the onshore and longshore components); Q=(PQ) s the sedment transport flux (P and Q beng respectvely the onshore and longshore components); C τ s the dmensonless Chezy coeffcent (typcally of order 0 - ); λ=-p s the gran packng (typcally 0.5 λ 0.8) beng p the beach porosty; g s the gravty acceleraton; subscrpts represent partal dfferentaton. The geometrcal terms presented n the prevous equatons are sketched n Fgure 3 here a typcal seabed evoluton s shon. An operatonal splt soluton of the NSWE/Exner system s acheved by separately solvng the NSWE and Exner equatons. The NSWE system s solved by usng the WAF method ell descrbed n..3 hose scheme s here labeled as H. The Exner system s solved by usng a fnte-volume method descrbed n.3. hose scheme s here labeled as M. The splttng scheme hch s a frst-order accurate n tme splt operator gves from the ntal soluton W n =(d n u n v n z n ) the soluton at the follong tme-step W n through the relaton: 6

36 n W = M ( H ( W n ). (.5) A detaled flo chart of the ntegraton procedure s provded n Fgure 4. In all the document superscrpt n represents the tme ndex hle represent space grd pont ndces referrng respectvely to the x and y coordnates. Fgure 3 Sketch of seabed profle and free surface elevaton... The hydrodynamc solver The hydrodynamc solver s that developed by Brocchn et al. (00). It uses a conservatve fnte-volume dscretzaton method to take nto account dscontnuous solutons (e.g. shock aves) by means of an approprate flux method. A TVD scheme s also appled by usng a flux lmter to avod spurous oscllatons due to dscontnuous solutons. 7

37 Fgure 4 Flo chart of the computatonal schemes.... System resoluton The vectoral conservatve form of the NSWE s: here ( U) G( U) S( U) U t F = (.6) x y 8

38 d U = ud F vd ud ( ) U U = u d gd = gu uvd U U UU U 3 (.7) vd U 3 0 UU 3 G( U) = uvd = ( ) = S U gdz C u U x τ v v d gd U 3 gdz C v gu y τ v U th z constant n tme. The system (.6) s splt to have to D-homogeneous equatons and a source equaton: ( U) = 0 U (.8) t F x ( U) = 0 U G (.9) t y ( U) U t = S. (.0) The global hydrodynamc scheme hch s a combnaton of to symmetrcal secondorder schemes can be descrbed as follos. If Hx s the scheme hch solves (.8) over the tme dt=t n t n Hy the scheme hch solves (.9) and So the scheme hch solves (.0) the numercal procedure that leads from the ntal condton U n to the soluton U n can be descrbed as follos: here H n ( H ( So( U ) So( H ( U ) n n U = n n ( U ) = ( Hx( Hy( U ) Hy( Hx( U ) n (.). (.)... Integraton: fnte-volume dscretzaton The rectangular doman S=[x -/ x / ] [t n t n ] n the x-t space s the fundamental unt used to dscretze the computatonal doman (see Fgure 5). An ntegral form of these equatons has been used over the rectangular volume S. For nstance equaton (.8) becomes: 9

39 = x / x U / [ U F( U) ] / t t n n ( x t ) dx U( x t ) dx F( x / t) dt F( x / t) dt = 0. ( A) x x S / ( B) t n n x ds = t ( C) n n t ( D) (.3) The cell averages can be defned as: n ( B) n ( A) n / ( D) n / ( C) U = U = F / = F / = (.4) x x t t here Δx = x / x -/ and Δt = t n t n. Substtuton of the averages nto (.3) gves: n ( / n F F / ) n n t U = U / /. (.5) x Fgure 5 Sketch of the doman S n the x t space...3. The WAF method At ths stage a sutable method s needed for representng the flux terms n (.5) and then for solvng the equaton. To have a good approxmaton even hen shocks occur the WAF (Weght Average Fluxes) method s used. The WAF method requres knoledge of the Remann structure of the equatons to be solved. 0

40 At the ntercell boundary x / and at tme level n e can defne a specal ntal value problem called sub-grd Remann problem th ntal condtons (U L U R ).. For equaton (.8) ths reads: U F U( x0) = U t 0 ( U) x = U U ( x) = U R L t AU = U = U n n x = 0 f x < x f x > x / / (.6) The characterstc curves of the soluton are determned through the egenvalues of the matrx A= F/ U then sutable multplers are found to decouple the set of PDEs of (.6) n a set of ODEs along hch Remann varables R propagate. In absence of source terms for the ODEs they propagate unaltered accordng to: dr dt k dx = 0 along = λk here k = 3 (.7) dt and are the Remann nvarants. Egenvalues are: (λ λ λ 3 ) = (u c u uc) th c = gd hle the Remann nvarants are (R R R 3 ) = (u c v uc). Hence the structure of the soluton of the specfc Remann problem reveals the presence of to knon constant states U L and U R and to unknon states U* L and U* R as also shon n Fgure 6. Values of U* L and U* R are taken as constant n agreement th the approach used by Toro (997). Hence neglectng the overbar sgn because varables are pece-se constant (thn each cell) the ntercell vector s defned as: L n n / U = U for t = t xa < x < x 0 A L n / U * for t = t x < < A x x n / A U / =. (.8) R n / U * for t = t xa < x < x A3 R n n / U = U for t = t xa < x < x 3 A4 The operatve value of the ntermedate flux F s estmated by the eghted average n / / (hence the name WAF) of the F(U) values of equaton (.8). Ths can be accomplshed by ntroducng the length of each segment cell dth Δx: β = A A l k k x Ak A th l=4. Hence: k appearng n Fgure 6 normalzed by the

41 x t x t x t x t = = = = λ β λ λ β λ λ β λ β (.9) Thus the follong eghted average fluxes are obtaned: ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ± ± = = 3 ) ( / / 4 3 / / * * l l l n n n n R L n n C F F U F U F F U F U F U F U F β β β β (.0) ΔF(l) beng the flux ump across the l th ave and C l = λ l Δt/Δx the Courant number of each ave. Fgure 6 Flux computaton from one specfc Remann problem n the x t space. Fnally generaton of spurous oscllatons caused by strong gradents s prevented by usng a Total Varaton Dmnshng (TVD) verson of equaton (.0) (see Harten 983). Ths s obtaned by means of the SUPERA flux lmter functon (Toro 997) / / ± n l f such that:

42 3 n n ( F( U ) F( U± ) sgn( Cl ) ( l) F F. (.) n / n / ± / = ± / fl l=..4. Shorelne treatment The treatment of the shorelne moton s really mportant to model properly the ntermttent flo caused by ave moton on a beach (e.g. see Brocchn et al. 00; Brocchn & Peregrne 996). In the present ork a ettng/dryng approach s used to predct the shorelne moton. The frst step conssts n the computaton of an equvalent ater depth for each cell by consderng the nflo volume that s the sum of all contrbutons occurred durng the prevous tme-steps. Then (Stoker 957) the WAF method s appled to the sub-grd Remann problem occurrng at the shorelne by evaluatng the equvalent depth n each cell as a functon of the ater level at the nodes around t. If the equvalent depth s larger than a threshold value d mn the cell becomes et. Durng the dryng phase f the ater depth s loer than d mn the cell becomes dry and the ater depth s set equal to zero. Snce the shorelne s the nterface beteen et and dry cells the shorelne boundary condton can be rtten as: dx dt s = s s v d = 0 (.) here x s s the poston of the nstantaneous shorelne v s and d s the flo velocty and the ater depth at the shorelne respectvely. The Remann problem that s consdered to solve (.) s defned rght at the edge beteen the last et grd pont x and the frst dry grd pont x. It s smlar to that modelng a movng pston (see Stoker 957) hch represents the shorelne advancng as shon n Fgure 7 and descrbed by: L ( U) = Ut AU x = 0 U( x0) = U f x < x / U F. (.3) t x Equatons for the moton of the shorelne are obtaned by determnng the value of u s for hch the constrant d s = 0 s satsfed. After some calculatons t can be obtaned u s = L L u L c L th c = gd. The smooth transton beteen the left state and the shorelne ~ can be descrbed by: ~ L L d u ] = [ d / ( u u ) / ] as shon n (Brocchn et al. 00). [ s 3

43 Fgure 7 Sketch of the soluton of the shorelne Remann problem and analogy th the movng pston. Fnally the flux at the shorelne boundary s: F L L ( d u ) ( λ λ ) t x ~ = λ F s. (.4) x n / t / F The expanson fan contrbuton s F ~ defned as: here λ =u L c L and λ s =u s. L L ~ ( d u ) F( d u ~ ) F( 0 u ) ~ F = F s (.5) The morphodynamc solver The morphodynamc solver has been developed to properly match the hydrodynamc solver hch provdes the forcng to update (.4) n tme. To be consstent th the hydrodynamc solver many of ts features have been employed n the morphodynamc resoluton e.g. the fnte-volume approach. 4

44 .3.. Sedment transport formulatons Before dscretzng the Exner equaton a sutable la must be chosen to reproduce the sedment transport. In the last decades a large number of formulas has been developed and s avalable n lterature. Because of the dfferent applcaton felds they have been found for some of them are very smple and depend on only a constant term that may vary over a rather de range (see for example the model of Kelly & Dodd 00 descrbed n the prevous secton). The choce of such a term s dffcult but t has been taken equal to that employed n some of the tests used for the code valdaton (see next chapter). Other formulas depend on several terms and are dffcult to be ntegrated n any solver. Such a dffculty s at the bass of the larger dffcultes n mplementng a fully-coupled model hch makes use of realstcally complete and complex sedment transport closures. Therefore more than one formulaton has been tested thn the nely-mplemented solver th the am to reproduce as ell as possble the valdaton tests descrbed n Chapter 4 hch requres the applcaton of dfferent transport las. As typcally done n all morphodynamc solvers the sedment transport (Q) s computed as the sum of to contrbutons: the bed-load transport (Qb) hch takes nto account the near-bed transport mechansms and the suspended-load transport (Qs) hch takes nto account sedment transport mechansms and partcle motons occurrng sgnfcantly far from the seabed. Hence t gves: Q = Qb Qs. (.6) Varous bed-load transport formulas can be used n dependence of the am of the computaton. They are presented n the follong. Qb = A v v (.7) beng the easest formulaton that has been tested. It depends on to terms only: the flo velocty v and the constant A that n turn depends on the sedment characterstcs and s assgned as an nput value. Ths formulaton s typcally used n fully-coupled hydromorphodynamc systems because even f t does not take nto account all the fundamental mechansms of sedment transport t s smple to ntegrate (see for example Grass 98). Such a formula has been used to valdate the solver th respect to the tests performed by Kelly (009) and Kelly & Dodd (00). One second bed-load transport formula s: Qb = Ad v v (.8) here A s a constant dependng on the sedment characterstcs v the velocty and d the total ater depth. The ntegraton of such a formula s a bt more complcated than the prevous one and as used by Kelly (009) hose tests have been reproduced for the valdaton of the solver. 5

45 ( θ γ z θ ) s z b = ρ 3 3/ θ γ Q 8 gd 50 c (.9) ρ θ γ z here ρ s the ater densty ρ s the sedment densty d 50 the medan sedment dameter γ a dmensonless number measurng the bed slope effects θ the Shelds parameter defned as ρ v v θ = Cτ ( ρs ρ ) and θ gd c the crtcal Shelds parameter C τ beng the Chezy frcton 50 coeffcent (see for example Beso et al. 003). Ths formula takes nto due account many mechansms and enables to consder the man sold transport phenomena related to the ntaton of sedment moton ncludng the stablzng effects due to gravty occurrng at the bottom. Ths s probably one of the best las for the evaluaton of the bed-load transport because hen large seabed gradents exst (e.g. n the presence of ether a submerged or an emerged bar) the stablzng gravty effect can be very mportant n the seabed reshapng. The suspended-load transport has been modeled only on the bass of the formula by Camenen & Larson (008): ε = sd Q s vcr exp (.30) s ε here s s the partcle settlng velocty ε = kd( D / ρ) the sedment dffusvty and D the total effectve dsspaton. The sedment concentraton c R s evaluated by means of to closure las. The frst one s that of Camenen & Larson (008): / 3 θ = M c R A cr θt exp 4. 5 (.3) θc here θ T and θ M are the mean and the maxmum shear stresses respectvely θ c s the crtcal shear stress and A cr a parameter dependng on the sedment dmenson (Camenen & Larson 008). The formula s rather complcated because of the large varaton n the evaluaton of each term n dependence of the ave/current characterstcs hence only the ave contrbuton has been taken nto account. The second formula s that of Van Rn (984): γ beng = d [( ) ] / 3 50 gυ * γ Ta * cr = 0.05d d (.3) s d and a ( b c ) c T = τ τ / τ here τ b s the bottom shear stress and τ c the crtcal shear stress (Van Rn 984). Ths easer formula has been tested and results compared th those of the prevous one. 6

46 .3.. System resoluton The Exner equaton s a scalar conservaton equaton: z t Q = 0 (.33) λ th Q beng the sedment flux dependng on z and on the flo varables uv and d hch are taken as constant durng each morphologcal tme-step. The eak form of equaton (.33) s obtaned by ntegratng t over the rectangular doman S = [x -/ x / ] [y -/ y / ] [t n t n ] n the x y t space and by usng the Stokes theorem: S z t Q ds = λ [ zdxdy Qdydt Pdxdt] = 0 Hence ntegraton of (.34) along the lne boundng the doman S gves: (.34) x / y / x / ( A) n t n n t y / y / y / x / x / / y n ( ) ( ) / n x y t dydx z x y t ( x y t) dydt / Q( x y t) / t n z Q P ( C) x ( x y ) / / t dxdt P( x y / t) dxdt = 0. t ( E) x y / x / ( B) n t n t y y / x / / ( D) ( F ) dydx dydt t n n t (.35) No t s possble to defne the follong averages as done for the hydrodynamc solver n equaton (.4): n ( B) n / ( D) n / ( F) Z = H / = J / = x y t y t x (.36) n ( A) n / ( C) n / ( E) Z = H / = J / = x y t y t x here x = x / x / y = y / y / and t = t Substtuton of such averages nto (.35) gves: n t n ( ) ( ) / n / t n / n H H J J / n n t Z = Z / / / /. (.37) x y n. 7

47 .3.3. Flux computaton In order to properly evaluate the fluxes appearng n (.37) the applcaton of the WAF method already appled to the hydrodynamc solver (see..3) has been chosen agan. Because of the structure of the Exner equaton Lax-Fredrch schemes or other schemes typcally used for hyperbolc equatons cannot be used. Further TVD schemes cannot be appled because of the non-homogenety of the Exner fluxes. Hence varous second-order schemes such as the WAF method tself (employed for the hydrodynamc flux evaluaton) have been tested. Hoever gven that the tme ntegraton of the WAF method s comparable to the mdpont second-order Runge-Kutta scheme and that the Exner fluxes do not sgnfcantly depend on the seabed elevaton a frst-order explct Euler scheme has been used. Fnally the fluxes are defned as: n n n ( Q( Z ) Q( Z ) n n / H ± / = U± ± U (.38) ( P( Z ) P( Z ) n / n n n n J ± / = U ± ± U. (.39).3.4. Shorelne treatment The shorelne s characterzed by the nterfaces beteen dry cells and et cells. The prevous fluxes are used for the et cells. The Exner fluxes at dry cells are equal to zero because there are no transport phenomena n such cells. Hence f cells ( ) and ( ) are respectvely et and dry the flux along the x-drecton at tme-step n/ and at x ±/ s: n n ( Q( Z ) 0) n / H ± / = U (.40).4. Numercal nstabltes and flterng Tme-steppng numercal models lke the present one can develop grd pont nose n the doman especally hen shock aves occur. Ths can be mtgated by usng a numercal flter. In partcular a lo-pass flter and a targeted flter can be used but at the expense of addng an undesrably hgh degree of vscosty to the model. Such an unanted effect can be mnmzed by applyng the flter only hen and here oscllatons are large. Indeed a common ay to obtan a good numercal stablty s to add some dffusvty nto the model that s useful for the numercal convergence too. In general spurous oscllatons are essentally due to the seabed treatment. In fact the resoluton of the hydrodynamcs makes 8

48 9 use of the SUPERA flux lmter to avod spurous oscllatons of the free surface hle the structure of the Exner equaton does not allo the employment of such a flter. Therefore z s the only varable to be nterested by the flterng. Actually none of the flters have been used for the smulatons descrbed n Chapter 4 but only used n some partcular cases n order to test ther applcablty..4.. A lo-pass flter: Shapro flter The Shapro flter (Shapro 975) belongs to the class of lo-pass dgtal flters typcally used n atmospherc/oceanc models to remove grd pont noses (see Veeramony & Svendsen 000). In order to obtan smoothng of the most unstable regons of the seabed profle thout ntroducng sgnfcant dampng of the nterested component (.e. z) a second-order Shapro flter has been chosen. Here belo the applcaton to a general functon at the grd pont s presented: ~ = = =. (.4) For the present D problem the flter has been modfed as: ~ = (.4) Ths flter s a hghly-dffusve flter thus t s appled only every p=0 tme-steps..4.. A second lo-pass flter: the targeted flter Ths s another lo-pass flter characterzed by non-unform multplers that are evaluated along x and y and affect only regons here oscllatons occur. The form of such a flter s:

49 30 ~ = Ay Ay Ay Ay Ay Ay Ay Ax Ax Ax Ax Ax Ax Ax (.43) here: ( )( ) ( )( ) < = 0 f 0 0 f 0 Ax Ax ( )( ) ( )( ) < = 0 f 0 0 f 0 Ay Ay Ax 0 and Ay 0 are numercal parameters representng respectvely the flter ampltude along x and y The stablty condton for ths flter s: Ax 0 Ay 0 /.

50 Chapter 3. The expermental tests Beach eroson n the sash zone occurs as a consequence of ntense sea storms even f coastal orks lke detached breakaters both emerged and submerged are used to defend the coast. Detached breakaters are the typcal defense structures characterzng the Italan coastlne n partcular the Adratc coasts. The man consequences of usng such barrers are: formaton of tombolos and salents dondrft eroson especally durng storm events localzed scours generaton of undesred rp currents etc. To better understand some of the phenomena nduced by dfferent types of breakaters and to desgn an optmal defense that may gve both good protecton and mnmum envronmental mpact a seres of expermental tests on a D physcal model has been carred out at the Hydraulcs Laboratory of the Unverstà Poltecnca delle Marche (Ancona). In a recent ork (Postacchn et al. 00) results comng from such laboratory campagns have been llustrated and a comparatve analyss of the seabed evoluton under the varous ave condtons has been performed payng partcular attenton to the short-term morphologcal changes of the beach as functon of the dfferent defense confguratons. Whereas many orks have been undertaken to characterze the effcency of submerged breakaters (e.g. Van der Meer 99 Buccno & Calabrese 007) only some have been performed to assess the overall hydrodynamcs functonng of the same structures and the nduced ater qualty (e.g. Lorenzon et al. 005) and even less studes have focused on the short-term evoluton of the cross-shore beach profles. In the follong sectons the expermental set-up and some results are dscussed hle the theoretcal approach and the fundamental features related th the crculaton occurrng around submerged and emerged breakaters are treated n Appendx A. 3.. The expermental set up The expermental tests have been performed at the Hydraulcs Laboratory of the Dpartmento I.S.A.C. of the Unverstà Poltecnca delle Marche (Ancona) that hosts a ave channel equpped th a avemaker for martme expermental models n reduced 3

51 scale. The flume s 50m long m de and.3m deep and can ork th a maxmum ater depth of m. The lateral flume alls here steel vertcal rods and glass ndos alternate for the central 36m enable to observe photograph and vdeo-record the ncomng aves the flos and the morphodynamc evoluton of the sandy bed. The ave generaton system provded by Wallngford (UK) s made of a vertcal paddle th a pston-type moton (maxmum stroke: m) actvated by an electrcal engne th poer of 4kW. The laboratory equpment ncludes an ADV employed to record the local flo velocty and 8 electro-senstve ave gauges that have been used to measure the ater levels as ell as all the ave characterstcs such as heghts perods etc along the flume (some of the ave gauges are sketched n Fgure 9). The model under analyss represents a cross-shore secton of the Gabcce Mare beach (PU Italy) a typcal seasde resort of the Italan East coast. As explaned n Lorenzon et al. (009) the reproduced secton s one of the most sgnfcant of such a lttoral area because t s characterzed by the breakater closest to the shorelne th the deepest salent thus ts beach s the most dffcult to be supported and the most crtcal to assess the postonng and most sutable shape of the defense barrer. Fgure 8 left panel shos the results of a bathymetrc survey of the hole Gabcce Mare beach carred out n September 008 by means of the multbeam method. The chosen beach cross-shore secton dentfed by a dashed lne as frst partally rectfed (to make a smpler model realzaton) then t as reproduced n the flume as a physcal model th a reduced scale of :0 (Fgure 8 rght panel). Also a submerged hump has been reproduced n the flume to represent the rocky structures characterzed by an algned dsplacement that s nclned th respect to the shorelne and located n the offshore area (dentfed by a blue crcle n Fgure 8 left panel). Fgure 8 Bathymetrc map (left) and reproduced movable-bed model n the flume (rght). Froude smlarty has been used for reproducng the hydrodynamcs hle the grans of the movable bed (d 50 =0.3mm) have been chosen accordng to the Dean crteron to represent the natural sand of the prototype beach (d 50 =0.mm). As shon n Fgure 9 sx breakater confguratons have been tested: three emerged (A n dark green D n magenta and E n purple) and three submerged (B n red C n yello 3

52 and F n green). Furthermore a confguraton th no structures (G n bron) has been tested. The man characterstcs of the varous confguratons are descrbed n Table here X s the breakater dstance to the shore (n meters) R c the berm level th respect to the mean ater level (n mllmeters) B the berm length (n mllmeters) off the offshore slope and n the nshore slope. The medan dameter of the breakater stones s 0.085m correspondng to.7m at prototype scale. S7 S6 S5 000 max stroke ave generator shore G A B D E F C m..l. ave sand Fgure 9 Sketch of the flume: shorelne (left) breakaters at the tested postons (mddle) and ave generator (rght); S5 S6 and S7 gve the postons of three ave gauges. Confguraton X (m) R c (mm) B (mm) off n A :3 : B :3 : C :4 :3 D :3 : E :3 : F :3 : Table Man geometrcal characterstcs of the tested breakaters at model scale. The ave attacks reproduced ere those due to three sea storms (called OS OS and OS3) occurred n the Adratc Sea n and 004 respectvely. The aves approached the shore th almost shore-parallel fronts. The peak ave heghts ere very severe (about 5m at prototype scale). Every storm has been reproduced by mantanng a constant superelevaton of the mean ater level ( ) to smulate the tme-averaged value of the storm surge (40cm for OS and OS3 7cm for OS n prototype condtons). The choce to reproduce real sea storms.e. short-term events as done to analyze and try to understand the beach behavor durng very strong events that n the last years occurred even durng the summer season hence resultng very damagng for the tourstc facltes. The entre perod of the chronologcal evoluton of each tested storm has been dvded n four dfferent phases hence t has been reproduced by means of four dstnct JONSWAP spectra each characterzed by dfferent ave heghts and perods. Such stages have been reproduced separately and n successon to smulate the complete storm. As an example the temporal evoluton of sea storm OS s shon n black n Fgure 0 hle the four 33

53 rectfed phases are dran n blue. Input values for each phase of the tested storms.e. sgnfcant ave heght (H s ) and peak perod (T p ) are summarzed n Table. Those condtons have been reproduced by the ave generator at a depth of 39.4cm (correspondng to 7.48m at prototype scale). Also one regular ave (OR) has been reproduced n order to analyze the seabed response under a constant ave forcng (constant heght H and perod T). Fgure 0 Chronologcal evoluton of the sea-storm OS th ave characterstcs (H s T p drecton and duraton) of each phase at both prototype and model condtons. Wave Phase Type Duraton (hh:mm) H s (m) T p (s) (m) OS spectrum 03: OS spectrum 0: OS 3 spectrum 05: OS 4 spectrum 08: OS spectrum 04: OS spectrum 04: OS 3 spectrum 03: OS 4 spectrum 3: OS3 spectrum 00: OS3 spectrum 00: OS3 3 spectrum 04: OS3 4 spectrum 0: OR - regular 04: :43 Table Characterstcs of the reproduced aves at model scale. 34

54 3.. Salent morphologcal results The cross-shore beach profle obtaned for each tested sea storm evolved gradually durng ts total duraton and as functon of the type of structure n use. In fact the transmsson coeffcent defned as the rato K t = H t / H here H and H t are respectvely the ncdent and transmtted (nshore of the barrer) sgnfcant ave heghts s dfferent for submerged and emerged structures. The transmsson provded by the submerged structures as K t 50-60% hle that gven by the emerged structures K t 40%. Because of ths the seabed has been surveyed at the end of each phase. In ths ay specfc phenomena lke the progressve stll-ater shorelne retreat the formaton of an emerged bar n the sash zone and varous scourng phenomena occurrng around the breakater locatons have been nvestgated Features of the seabed profles The superposton of all the bed profles obtaned for confguraton D after each phase of OS3 s plotted n Fgure. Ths ell represents the general trend of the profle evoluton durng any sea storm and for any confguraton. The man features of the seabed evoluton are: an emerged berm at the nshore end of the beach (on the left) a sgnfcant shorelne retreat some sgnfcant scourng at the offshore toe of the breakater OS3. OS3. OS3.3 OS3.4 Intal profle Fgure Superposton of the bed profles after each phase of the sea storm OS3 (September 004) for confguraton D. 35

55 A large groth of the berm durng the to fnal phases of the storm (OS3.3 n purple and OS3.4 n cyan) s evdent hereas the peak phase (OS3. n green) forces both an mportant shorelne retreat and most of the fnal scourng on the offshore sde of the breakater. A steepenng of the shalloest porton of the beach profle s only evdent durng OS3.3 and OS3.4 that are the longest phases and therefore the most reshapng ones. The emerged berm at the sash zone s very large and ts dmensons are larger than the berm expected on a real beach durng the summer perod probably because of dstorton problems affectng the sedment scalng. Ths conecture seems to be partally confrmed by the rather large rpple formaton all over the movable bed Seabed evoluton In order to apprecate the nfluence of to mportant parameters n the dsspaton nduced by detached breakaters such as the dstance to the shore X and the berm level R c the morphodynamc response of the dfferent confguratons to the varous storms have been compared. Fgure shos the fnal profles produced by OS n the presence of to emerged (top panel confguratons A and D n green and magenta respectvely) and to submerged breakaters (bottom panel confguratons B and C n red and yello respectvely) th respect to the ntal profle (black lne) that as re-shaped by hand at the begnnng of each storm. If e take as the ntal shorelne the ntersecton beteen the ntal profle (dashed lne n Fgure ) and the blue lne representng the mean ater level ( m..l. n Fgure ) then the fnal shorelne s the ntersecton beteen the fnal profle (colored sold lnes n Fgure ) and the m..l. Hence the shorelne retreat s represented by the dfference beteen ntal and fnal shorelnes. Observng Fgure t s clear that both types of breakaters nduce sgnfcant retreats but globally submerged breakaters nduce some retreat of the stll-ater shorelne (less than m) hle the emerged barrers provde larger retreats (more than.5m). At the end of the run (regardless of the type of forcng) an emerged bar formed n the sash zone hose sze as larger hen submerged barrers ere used (confguratons B and C). Therefore the ave transmsson provded by submerged barrers s larger than that nduced by emerged structures so that the submerged breakaters nduce a more evdent reshapng of the upper part of the beach and a steepenng of the sash zone by nducng a localzed eroson and a large emerged berm. In the presence of emerged structures the sash zone occurs over a mldly slopng beach and less sand s moblzed. The man cause of such a dfference s attrbuted to the ntense seaard sedment transport occurrng n the freeboard regon over the berm of submerged breakaters. Hence for confguratons B and C the sand transport toards the offshore s much more mportant. Ths behavor s partcularly evdent after storm OS as shon n Fgure (top panel). 36

56 00 50 m..l. surge (36mm) 0 m..l Intal profle Confguraton A Confguraton D m..l. surge (36mm) 0 m..l Intal profle Confguraton B Confguraton C Fgure. Comparson among the fnal profles due to OS for emerged confguratons A and D (green and magenta lne top panel) and for submerged confguratons B and C (red and yello lne bottom panel); the black dotted lne represents the ntal profle; horzontal and vertcal scales are n mllmeters. The dstance from the shore has a mnmal nfluence on the eroson extent. Probably ths behavor s strctly related th the dynamcs occurrng n the regon beteen the structures and the shorelne the aves there undergo a very lmted shoalng. The dfferent eroson extent beteen emerged and submerged structures depends on the dfference n the transmtted aves but seems to depend more on the ater plng-up. For the submerged 37

57 confguratons the larger plng-up nduces breakng to occur closer to the shore hence gvng a larger shorelne retreat (due to both the underto and the sash moton). 38

58 Chapter 4. Valdaton of the numercal solver The present chapter s devoted to llustrate a seres of smulatons amed at testng the capabltes of the hydro-morphodynamc solver n properly reproducng the typcal hydromorphologcal characterstcs. The analyss has been performed by comparng the numercal results th solutons provded by other numercal solvers by analytcal models and by expermental tests. In the follong sectons ( 4. and 4.) to seres of dfferent tests characterzed by a seabed morphologcal evoluton nduced by bed-load transport are presented. Both of them start from a dam-break event and the flo propagates through the doman provdng sgnfcant modfcatons of the movable seabed. In partcular n the typcal dam-break problem a shock ave propagates on a dry regon. A gate located at x = 0 dvdes to sem-nfnte spatal regons: on the left of the gate the ater stays at a fxed level h 0 hle on the rght there s no ater. The gate s removed at tme t = 0 and the ater flos n the doman. The sh to valdate by means of these tests to formulas (.e. equatons (.7) and (.8)) that have been presented for the descrpton of the bed-load transport leads to perform several smulatons characterzed by dfferent geometres sedment propertes ntal and boundary condtons. The frst set of smulatons ams at reproducng the tests performed by Fraccarollo & Capart (00) by means of ther model descrbed n.3.4. They reproduced some small-scale laboratory dam-break aves characterzed by an ntal ater depth h 0 = 0cm that s released over erodble beds n a prsmatc channel. The second set of smulatons refers to the dam-break tests carred out by Kelly (009) n order to estmate the rate of change of the morphologcal evolutons nduced by dfferent bed-evoluton coeffcents A. The model has been descrbed n detal n.3. and.3.. The sedment transport only due to suspenson has been nvestgated as llustrated n 4.3 here t s shon that the hydro-morphodynamc solver has been valdated by means of the analytcal solutons of Prtchard & Hogg (005). The hydrodynamc problem that has been studed n the paper at hand can be seen as ether a dam-break or a bore event and provdes the same soluton as suggested by Shen & Meyer (963). From the morphodynamc pont of ve Prtchard & Hogg (005) apply the bore approach and analyze the sedment 39

59 transport rates nduced by a sngle uprush-backash event over a planar nclned beach. The analytcal decoupled model by Prtchard & Hogg (005) has been descrbed n.3.3. Other tests am to check f numercs and experments lead to smlar results or f numercal approxmaton and laboratory/scale effects make the solutons totally dfferent. The reference laboratory experences have been carred out n the ave flume of the Unverstà Poltecnca delle Marche and are descrbed n Chapter Valdaton of the bed-load transport: Fraccarollo & Capart (00) Here belo the numercal tests performed by Fraccarollo & Capart (00) FC0 n the follong are analyzed. FC0 reproduced numercally to expermental sets that dffer prmarly n the used sedment materal and n the locaton they have been carred out. The frst campagn has been performed n Tape (Natonal Taan Unversty) the second n Louvan-la-Neuve (Unversté Catholque de Louvan). In the Tape expermental tests a very lght sedment as used. The grans ere artfcal pearls covered th a shny hte coatng sphercal n shape havng a dameter of 6.mm densty ρ s = 048kg/m 3 and fall velocty s = 7cm/s. The correspondng densty rato s = ρ s / ρ =.048 makes the materal slghtly heaver than ater. For the tests performed usng a horzontal prsmatc flume of rectangular cross-secton at Louvan-la-Neuve denser sedments ere chosen. These ere characterzed by an equvalent dameter of 3.5mm a densty ρ s = 540kg/m 3 (gvng a dmensonless densty s =.54) and fall velocty s = 8cm/s. Before each test seres sedments ere arranged n the flume and profled n order to have an ntal confguraton of constant thckness of about 5-6cm. A vertcal gate separated to regons of the flume that ere flled th ater. Donstream of the gate the sedments ere saturated the ater reachng the maxmum level of the granular bed. Upstream the stll ater level rased up to h 0 = 0cm over the maxmum level of the granular bed. The gate removal as very rapd and led to the ntaton of a dam-break event lastng very fe seconds. A dgtal partcle trackng velocmetry (DPTV) as used to acqure the gran motons and to vsualze both transport pattern and nstantaneous morphologcal changes. The Tape measurements refer to a tme range equal to t = 0-5t 0 here t 0 = (h 0 /g) t/ s the hydrodynamc tme scale. The tme cutoff t = 5t 0 depends on the valdty of the shalloater assumpton (t»t 0 ) but because of the very lght sedment used (s =.048) the tme requred for the flo to equlbrate ts sedment load s larger than 5t 0 and non-equlbrum sedment transport effects cannot be neglected. Hoever the hgh sedment moblty makes the transport process very nterestng llumnatng a number of features of the phenomena and yeldng to nterestng results. 40

60 Because of the mentoned lmtatons the Louvan tests have been conducted. In ths case the tme range s der than before (t = 0-0t 0 ) and t s more consstent th the expected doman of valdty of the theory n use. The man processes that are nvolved pertan to the ater surge that s released by the gate. It quckly entrans bed sedment n ts moton and an mmedate eroson s most ntense n the vcnty of the dam th a scour developng both upstream and donstream of the ntal gate poston. The bulkng of such a surge th eroded materal s most severe for the lght sedment materal (Tape tests). Fgure 3 Snapshots from the expermental tests of Tape at t = 0s 0.s 0.s 0.3s 0.4s 0.5s (left panels from the top to the bottom) and Louvan at t = 0.5s 0.50s 0.75s.00s (rght panels from the top to the bottom adapted from FC0). An mportant feature n both seres of tests s the heap of sedments at the front of the ave. In fact the donstream bed s sept nto the flo as the bore advances. The Tape lght-sedment experments are characterzed by an mportant flo feature absent n FC0 s theoretcal descrpton: the scour hole endurng near the gate poston hch provdes a non-monotonc free-surface profle some near-surface turbulence and rregular aves radatng energy donstream (see Fgure 3 left panels). 4

61 The Louvan experments performed th heaver grans s also characterzed by a localzed scour nearby the ntal hydraulc ump poston because of the large dsturbance nduced by the sudden gate removal but th a loer degree (Fgure 3 rght panels). In order to check the assumpton of contact-load transport FC0 verfed that flo condtons used for both sets of experments remaned beneath the suspenson threshold.e. u * < here u* s the frcton velocty and f the partcle fall velocty. f 4... Numercal results from the hydro-morphodynamc (HM) solver The comparson beteen the expermental results above presented and shon n Fgure 3 and numercal smulatons performed by means of the model (.3.4) of FC0 leads to the analyss of some features such as the lmt of the transport layer Γ s and the behavor of the mxture layer that cannot be nvestgated by the hydro-morphodynamc (HM) solver descrbed n the present ork. In fact as hghlghted n.3.4 the HM solver enables to evaluate the seabed varatons nduced by the bed-load transport meanng all that s n contact th the seabed and by the suspended-load.e. all that s advected by the flo nsde the ater column. No separaton lmts are descrbed n the model and no dstncton beteen pure clean ater and lqud-granular mxture are done. All the descrbed dam-break solutons are shon together n Fgure 4 and Fgure 5 llustratng expermental and theoretcal results related to the Tape and Louvan respectvely. For the former test the output that s obtaned after t = 0.5s s shon hle for the Louvan test the fnal consdered tme s t = s n order to respect the tme lmtaton descrbed n 4.. For hat concerns the smulatons performed by means of the HM solver the tme step employed for both tests s 0.s hle the test duraton s 0.5s for Tape and s for Louvan smulatons as n FC0 s tests. For hat concerns the spatal dscretzaton the doman s descrbed by the follong mnmum and maxmum values along x and y: x mn = -0.6m x max = m y mn = 0 and y max = 0.m. The dstances beteen each par of nodes are x = 0.00m and y = 0.004m n the x and y drecton respectvely. Morphologcal characterstcs such as densty and dameter of sedments are the same as n the laboratory experments descrbed n FC0 hle the Chezy coeffcent s C τ = About sedment moton to represent correctly the sheet flo transport only bed-load transport has been consdered hence suspended-load contrbuton has not been taken nto account. In fact sheet flo (see also.) can be consdered as a nearbed transport occurrng at a slghtly hgher level and s much more correlated th bed-load than th suspended-load transport. In partcular equaton (.8) has been adopted th a coeffcent A = 0.004s m -. 4

62 Fgure 4 Results from Tape tests after t = 0.5s : black lnes gve the expermental results green lnes the results by FC0 red lnes numercal solutons of the HM solver; sold lnes refer to the stll ater level Γ dash-dotted lnes to the Γ s lmt dashed lnes to the bottom level Γ b. Fgure 5 Results from Louvan tests after t = s : black lnes gve the expermental results green lnes the results by FC0 red lnes numercal solutons of the HM solver; sold lnes refer to the stll ater level Γ dash-dotted lnes to the Γ s lmt dashed lnes to the bottom level Γ b. 43

63 In Fgure 4 the results of the expermental tests performed n Tape are represented n black hle the soluton of FC0 s n green. In both cases sold lnes are the stll ater level Γ dash-dotted lnes the lqud-granular mxture boundary Γ s and dashed lnes the bottom level Γ b separatng the flong mass from the sold bed-rock. The numercal results found th the HM numercal solver are characterzed by to boundares only.e. those referrng to the ar-ater separaton (sold red lne) and to the bottom boundary (dashed red lne). Lke for the Tape test fgure Fgure 5 gves n black green and red respectvely the results found durng the small-scale experments performed n Louvan the results of FC0 and the HM numercal soluton. In both cases the expermental results match farly ell th the theoretcal solutons of FC0 for the moton of flo and mxture and for the seabed change. Hoever the expermental dam-break ave generates a rather ntense scour n correspondence of the gate poston hereas a smoother eroson occurs donstream and upstream of t. Comparson th the HM numercal soluton leads to a rather postve response. In fact though the scour ntensty and the heap of the eroded sedments at the ave front are smaller than those predcted by FC0 the fnal shape of both stll ater level and seabed evoluton s qualtatvely smlar to that provded by the model of FC0. Wth reference to the expermental tests the HM soluton for the seabed locaton seems to be n beteen the bottom level Γ b and the lmt of the transport layer Γ s except for the ntense scour occurrng at the gate locaton (such a localzed eroson not captured even by FC0 s soluton). Moreover the ave speed seems to be a good compromse beteen the expermental one and that gven by FC0: the ave front of the HM soluton remans n beteen the ave fronts of the other to solutons. It s beleved that the dfferences n the predcton of the seabed change gven by the to models (HM and FC0) s partly due to the dfferent formulaton of the shallo-ater problem at hand. FC0 used a degree of detal larger than that typcally used n ths type of problems: as explaned n.3.4 they took nto account three dfferent flo layers (ntroducng a lqud-granular mxture) nstead of the typcal to layers. The HM model s smlarly to most of the nearshore morphologcal solvers bult on a to-layer approach (the NSWE gvng the hydrodynamc evoluton and the Exner equaton that for the seabed). Hence such an approxmaton leads to neglect the vertcal dstrbuton of the sedment transport. Furthermore the sedment transport formulas used n the HM account for bedload and suspended-load and dsregard possble contrbutons gven by the sheet-flo transport as recalled n the descrpton of the model of FC0 n.3.4. These fundamental dfferences can lead to an underestmate of the erosonal potental of a dam-break event over a bottom made of very lght spheres and hose duraton s very short. 44

64 4.. Valdaton of the bed-load transport: Kelly (009) Heren results of the valdaton of the HM solver s capabltes n modelng the bed-load transport are summarzed. In the specfc comparson s made th respect to the tests performed by Kelly (009) by means of the fully-coupled models descrbed n.3. (Kelly & Dodd 00) and.3. (Kelly 009). For ths purpose sedment transport due to suspenson has been deactvated as n the prevous valdaton tests of FC0 n order to ork th the same las used by Kelly (009). The solver developed for the sedment transport formulatons descrbed by (.7) and (.8) has been run th a number of dfferent values of the constant parameter A that s estmated on the bass of feld data through the formula: 3 A = 8 f ( ) R g s (4.) here s s the non-dmensonal sedment densty and f R the frcton coeffcent. The ntal condtons for all the tests are gven by: h d( x0) = 0 0 f x 0 f x > 0 and u( x0) = 0 x (4.) th h 0 = m. The porosty s zero n all the tested cases thus the parameter ξ (see equaton (.3)) s alays equal to one. Each test lasts 5s. Numercal smulatons of all the dam-break events llustrated n Kelly (009) KE09 herenafter have been carred out th the HM solver n order to valdate the code for hat concerns the use of both formulatons (.7) and (.8). The tme step employed for the tests s s hle the test duraton s of 5s. For the spatal dscretzaton the doman s descrbed by the follong mnmum and maxmum values along x and y: x mn = -0m x max = 3m y mn = 0 and y max = 0.4m. Nodes are placed at a dstance x = 0.005m and y = 0.m n the x and y drecton respectvely. The morphologcal characterstcs are a sedment densty ρ s = 048Kg/m 3 and a sedment dameter d 50 = 6.mm hle the Chezy coeffcent s C τ = 0.0. The coeffcent A s the same used n KE09 and s dfferent n each test as shon here belo. In the follong comparsons beteen results of the solvers at hand are llustrated. Each fgure refers to the spatal evoluton of dmensonless varables such as ater depth d horzontal velocty u and bottom level z at tme t = 5s here t = 0 s the begnnng of the dam-break event.e. the gate release. Such varables are made dmensonless by dvdng them by h 0 n the case of d and z and by (gh 0 ) / n the case of u hle the x coordnate s dvded by h 0 fnally tme s made dmensonless by means of the scale (h 0 /g) /. Moreover n the fgures hch follo blue lnes refer to the HM results hereas red lnes pertan to the KE09 s solutons. 45

65 In partcular the results llustrated n Fgure 6 Fgure 7 and Fgure 8 have been acheved by means of the smpler sedment transport closure that as used and mplemented by KE09.e. (.7) that also has been analyzed n Kelly & Dodd (00). Three dfferent values of A have been used.e. 0 3 s m (Fgure 6) s m (Fgure 7) and 0 s m (Fgure 8) so that the transport rate turns out to be larger for larger coeffcents. Fgure 9 and Fgure 0 refer to equaton (.8) and n ths case the dmensonal constant parameter A has been taken equal to.5 0 s m and s m respectvely. Fgure 6 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Au 3 here A = 0 3 s m. 46

66 Fgure 7 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Au 3 here A = s m. Fgure 8 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Au 3 here A = 0 s m. 47

67 Fgure 9 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Adu 3 here A =.5 0 s m. Fgure 0 Numercal results of KE09 s model (n red) and of the HM solver (n blue) at tme t = 5s after a dam-break event; from top to bottom: dmensonless depth velocty and bottom locaton for q = Adu 3 here A = s m. 48

68 By nspectng the global trends of the mentoned fgures t s found that the ncrease of the parameter A leads to sgnfcant effects on both velocty dstrbuton and seabed evoluton. Consstently th the expermental/theoretcal results of FC0 the maxmum seabed eroson occurs near the gate locaton (x = 0) hle the largest deposton s found donstream more or less at the bore-ave front (see 4.). In the comparson among the numercal results obtaned th closure la (.7) t s mportant to note that a larger coeffcent A provdes larger erosons both n depth and n streamse extenson also gvng a much more ntense accreton donstream n correspondence of the bore front here both horzontal velocty and bottom level reach ther maxmum. For the second bed-load equaton (.8) the eroson/deposton pattern s qualtatvely very dfferent from the prevous solutons. In fact the largest seabed elevaton s reached much upstream of the ave tp smoothly decreasng don to the front locaton. Also n ths case larger erosons and accretons are obtaned for larger coeffcents A. The results obtaned by means of the HM solver are farly close to the solutons of KE09 especally for the cases computed th closure (.8) and shon n Fgure 9 and Fgure 0. Here the blue and red lnes overlap almost perfectly for all the varables under analyss though KE09 s bore s faster than that computed by the HM solver n both Fgure 9 and Fgure 0. In fact nspecton of the velocty evoluton (mddle panels of both fgures) shos that red lnes cover a larger spatal range suggestng that the ave front reaches larger values of x. Fgure 6 Fgure 7 and Fgure 8 sho a farly good agreement beteen blue and red lnes some oscllatons characterze the HM s soluton at for the seabed elevaton near the front of the erosonal bore. It s beleved that these are an undesred outcome of the eak couplng beteen hydrodynamcs and morphodynamcs beyond possble naccuraces n the computaton of the shorelne moton as forced by shock aves. As already sad no flterng has been employed though the use of one of the flters descrbed n.4 ould smooth sgnfcantly the oscllatng regon thus provdng solutons that are more smlar to KE09 s results but at the same tme they ould represent further approxmatons of the phenomenon at hand. At the end a good agreement beteen the compared solvers can be seen though the HM solver s a decoupled model and provdes some spurous oscllatons hen strong dscontnutes are present Valdaton of the suspended-load transport: Prtchard & Hogg (005) A valdaton of the representaton of the suspended sedment transport has been carred out by reproducng the tests performed th the analytcal model of Prtchard & Hogg (005) descrbed n.3.3 and herenafter called PH05 and comparng ther results th those obtaned through the HM solver. 49

69 PH05 developed ther analytcal model n order to take nto account only suspended sedment transport. Further the model s able to account for sedment that s moblzed by the ntal bore and advected nto the sash zone as ell as the net transport hch arses from sedment settlng lag effects. An mportant scope of ther study as to nvestgate the contrbuton of these to mechansms to net sedment fluxes and therefore they started from the analyss of the base case n hch none of the to above-mentoned mechansms (ntal sedment concentraton and lag) occur. Account of these to mechansms ncluded n PH05 s model enables to evaluate a nonequlbrum suspended-load transport hereas hen neglectng them only a classcal steady-state transport can be descrbed. In the present thess e focus on the descrpton of the smpler steady-state transport.e. dsregardng both concentraton and delay mechansms hch have not been modeled by the current verson of the HM solver. Furthermore only suspended sedment transport has been consdered hle the bed-load contrbuton has been deactvated n the HM smulatons. The analyss has been carred out on the bass of the dmensonless varables used by PH05. Hence the hydrodynamc quanttes are scaled by means of the vertcal excurson of the sash A and of the natural concentraton scale C = m e / s. One mnor dfference dstngushes the to models: the HM solver makes use of a horzontal x-axs rather than beng bed-parallel as n PH05. For a clear dentfcaton of all varables dmensonless propertes are dentfed by a tlde (~) those referrng to the analytcal model by PH05 are labeled th a hat (^) hle those related to the HM solver are plan. Takng nto account that β s the nclnaton of the beach varables have been made dmensonless as follos: ~ xˆ sn β x tan β ~ x = = t = tˆsn β A A ~ hˆcos β h ~ uˆ h = = u = = A A ga ~ ˆ ~ ~~ huc ˆˆ cos β q = huc = = C 3 ga q c R g g = t tan β A A u ~ cˆ c = ga C ga 3 (4.3) here q and c R are respectvely the sedment transport rate and the reference concentraton found through the HM solver. The reference concentraton of the HM model (c R ) s taken as the scalng concentraton C used n the analytcal model. As explaned n.3.3 the tests at hand represent the dynamcs of the ell-knon problem of the flo follong a bore collapse on a planar nclned beach that has been studed by Shen & Meyer (963). Such a flo may be nterpreted as an exact soluton for ether an ntal value problem or a boundary value problem. In the ntal condton nterpretaton ~ the flud s ntally at rest th a unform dmensonless depth h = behnd a dam located at ~ x = 0 ; at ~ t = 0 the dam s removed and the flud s alloed to flo out and to accelerate donslope. In the boundary condton nterpretaton only the flo n the regon 50

70 close to the shorelne s consdered regardng the flo as drven by ncomng ~ characterstcs hch satsfy the condton u ~ h ~ t =. PH05 treated t as a boundary condton problem for hch the forcng s a bore approachng the beach and provdng an ntal uprush and a consequent backash. PH05 tested several suspended-load formulas th the am to understand the adaptaton of the man suspended-load las that are present n the lterature: thus they made qˆ to depend on more than one combnaton of û and ĥ. At the same tme formula (.30) proposed by Camenen & Larson (008) has been used n the HM solver for the computaton of the suspended-sedment transport. Such a formulaton can be regarded as a specfc case n hch the sedment transport rate depends on both cubc velocty and ater depth.e. q u 3 d; thus t can be traced back to one of the formulas employed by PH05. The tme step employed for the tests s 0.0s hle the test duraton s of 4s. For hat concerns the spatal dscretzaton the doman s descrbed by the follong mnmum and maxmum values along x and y: x mn = -0.m x max = 3.m y mn = 0 and y max = 0.m. The dstance of the consecutve nodes are x = 0.0m and y = 0.0m n the x and y drecton respectvely. The morphologcal characterstcs are: sedment densty ρ s = 650Kg/m 3 and sedment dameter d 50 = 0.08mm. Fgure and Fgure sho respectvely the analytcal solutons found by PH05 and the numercal results obtaned through the HM solver. Varables are presented n ther dmensonless form: x ~ t ~ Q ~ and q ~ are respectvely the horzontal drecton tme net flux and nstantaneous flux as defned n (4.3) and (.8). (a) (b) ~ ~ ~ ~ Fgure PH05 soluton for the net fluxes Q ~ (panel a) and the nstantaneous flux q ~ (panel b): (a) dashed lnes gve the net fluxes occurrng durng the uprush (postve values) and the backash (negatve values) the sold lne gves the net flux over a cycle; (b) sold lnes gve the flux at samplng ponts located at x ~ = (adapted from PH05). 5

71 The analytcal results llustrated n Fgure a underlne the asymmetrcal contrbuton of the uprush (dashed lne n the postve range) and backash (dashed lne n the negatve range) to the suspended-sedment transport occurrng over the hole sash zone. The negatve flux developng durng the backash s predomnant hence the global net flux durng the entre sash cycle (sold lne) s offshore drected and the bed evoluton s thus affected by an eroson that s much more ntense than the sedment deposton. Fgure b compares the temporal evoluton of the nstantaneous flux q ~ at four dfferent locatons ~ along the horzontal doman correspondent to the dmensonless coordnates: x = The hghest curve refers to the pont that s the more offshore.e. ~ x = 0.5 hle the loest pertans to the more onshore ~ x =. The latter corresponds to the mddle pont of the sash zone because follong (4.3) x = A/tanβ that s half of the maxmum horzontal excurson beng A the vertcal excurson of the sash. It s nterestng to note that the dfference n the transport rate at the extreme samplng ponts (.e. ~ x = 0.5 and ~ x = ) s very large. In fact the maxmum flux at ~ x = 0.5 s about 0 tmes the peak value at ~ x =. In Fgure results are llustrated as obtaned by usng thn the HM solver three dfferent frcton coeffcents (C τ = and 0.00) varable that n PH05 s not ell defned. By analyzng the temporal evoluton of the nstantaneous flux comng from use of dfferent coeffcents (panels b d f) a very smlar trend s observed even f the soluton covers an ncreasngly large streamse extent th ncreasng values of the frcton coeffcent. Further by usng a small (C τ = 0.00) or a large coeffcent (C τ = 0.0) the same trend of spatal evoluton of Q ~ s obtaned though the net flux covers a der range hen larger frcton coeffcents are adopted. The net flux Q ~ (left column of Fgure ) s more ntense n absolute value durng the uprush than durng the backash hence provdng an onshore-drected transport at the end of the entre sash cycle. The clam of Kelly & Dodd (00) also reported n.3.3 that decoupled models cannot reproduce near-shorelne beach accreton seems to be nvald for the eakly-coupled HM model. Actually the net fluxes predcted by the HM solver seem to be strctly correlated th the frcton coeffcent and the total flux decreases th C τ. By nspectng panels a c and e the negatve flux evoluton referrng to the backash transport remans alays the same and s consstent th the net flux evoluton llustrated by PH05 (loer dashed lne of Fgure a). On the other hand the uprush contrbuton reduces hen the beach frcton decreases but t s alays larger than that of PH05 s soluton (upper dashed lne of Fgure a). Hence use of an almost vanshng frcton may yeld to results that are more consstent th PH05 s soluton. Hoever the smulaton run th a coeffcent C τ = 0 gave poor results because numercal errors such as spurous oscllatons on the seabed profle occurred. 5

72 (a) (b) (c) (d) (e) (f) Fgure Spatal evoluton of Q ~ (panels a c e) and temporal evoluton of q ~ (panels b d f) from the HM solver obtaned by usng a frcton coeffcent C τ = 0.00 (top panels) C τ = (mddle panels) and C τ = 0.0 (bottom panels). As already mentoned scales are largely dfferent but the trend of the sngle curves shon n the rght panels of Fgure are qualtatvely smlar: they represent the nstantaneous 53

73 flux evoluton n correspondence of the ponts that are located at ~ x = 0.5 (purple lne) ~ x = 0.5 (red lne) ~ x = 0.75 (green lne) and ~ x = (blue lne). If e consder the hghest and the loest curves the maxmum nstantaneous flux occurrng at ~ x = 0.5 (purple lnes) s slghtly more than the double of the maxmum q ~ occurrng at ~ x = (blue lnes). In other ords the maxmum fluxes q ~ at ~ x = 0.5 and ~ x = are characterzed by peaks that are n a rato of about to hle n PH05 s soluton the rato beteen these to maxmum values s of about 0 to. In summary the HM model predcts a suspended sedment transport that s more ntense n both onshore and offshore drectons n correspondence of beach postons that are closer to the maxmum sash excurson th respect to results provded by PH05 s model. In addton hle the backash flux s not affected by frcton coeffcent changes the transport rate occurrng durng the uprush depends strongly on C τ. Furthermore accordng to the HM solver s results the total transport s alays onshore drected after a sash cycle hle PH05 s soluton predcts an offshore drected transport as shon n Valdaton of the total transport: flume experments As sad n Chapter 3 some expermental tests characterzed by a sandy bed and several breakater confguratons have been carred out n the Hydraulcs Laboratory of the Unverstà Poltecnca delle Marche (Ancona). Wth the am to valdate the HM solver by means of expermental data some of the aves that ere run and some of the confguratons that ere bult n the ave flume have been reproduced numercally. Fnally the morphologcal evoluton of the beach provded by the numercal solver has been compared th that observed n the laboratory. Among all the aves reproduced n laboratory (see Table ) to have been smulated th the HM solver. A tme seres of the ater elevaton has been used to represent the offshore boundary condton for the numercal solver. It has been taken from the measurements performed by means of one of the ave gauges used n the expermental tests (see 3.). More precsely the tme seres collected by probe S4 that as the farthest from the ntal shorelne among the probes located over the sandy seabed has been used. S4 as placed about 4m from the ntal shore poston (see Fgure 9 here shorelne and postons of S5 S6 and S7 are shon). In order to test the HM solver th ave forcng of dfferent nature.e. both regular and rregular to aves have been run n the numercal smulatons: an rregular JONSWAP ave OS phase herenafter called OS and the only regular ave OR reproduced n the flume (for more detals see Table of Chapter 3). The choce of OS s due to the need of reproducng the frst phase of one of the sea-storms that ere run n the flume. In fact the ntal bathymetry.e. that one before each storm as free from to-dmensonal 54

74 effects of the seabed hch nstead ere common at the end of each reproduced phase because the sandy bottom as reshaped at the end of each storm accordng to a elldefned ntal profle. To confguratons chosen from those reproduced n the flume (see Table ) have been smulated through the HM solver: confguraton G thout no defense structures and confguraton B characterzed by the submerged breakater that s the closest to the beach among all the tested barrers. Confguraton G has been smulated by means of the HM solver that s descrbed n Chapter. Such a solver s able to ork th ust one type of sol th specfc characterstcs lke sedment dameter and densty over the entre doman hence n the case of confguraton G the only materal s a homogeneous sand. On the contrary representaton of confguraton B cannot be done through the prevously-descrbed solver because of the submerged breakater present nsde the doman. It s characterzed by a dfferent materal.e. rock th respect to the movable seabed and dfferently from the sand t s not largely affected by the ater moton. Hence an extenson of the HM model has been done n order to take nto account the presence of the barrer. In partcular the possblty to nclude a rgd body nsde the doman.e. the submerged structure n the case at hand has been mplemented. The rgd structure s modeled as an mpermeable element here no eroson can occur hle deposton s alloed for (for more detals see Appendx B). Errors assocated th such an approxmaton can derve from the omsson of nfltraton/exfltraton flos through the rocks that actually represent mportant contrbutons n the crculaton pattern occurrng close to rubble-mound breakaters. The closure las that have been used n the numercal smulatons are: equaton (.9) for the bed-load transport Qb that takes nto account the ntaton of sedment moton ncludng the stablzng effects due to gravty occurrng at the bottom; equatons (.30) and (.3) for the suspended-load Qs due to Camenen & Larson (008). Referrng to the numercal smulatons of both confguratons the output tme step for all the tests s 000s hle the test duraton s of 7000s and 000s n the case of respectvely regular ave and spectral ave. For hat concerns the spatal dscretzaton the doman s descrbed by the follong mnmum and maxmum values along x and y: x mn = -0.05m x max = 9m y mn = 0 and y max = m. The horzontal dstance along x beteen to contguous nodes s x = 0.08m from x = 0 and x = m and x = 0.0m n the range x = -9m. In the y drecton y = 0.08m. The sedment densty s ρ s = 500Kg/m 3 the sedment dameter d 50 = 0.5mm and the frcton coeffcent s C τ = 0.0. In the next sectons results of both tests (th confguratons G and B) are llustrated startng from the analyss of the man features characterzng numercal results and fnshng th the comparson beteen expermental and numercal results. 55

75 4.4.. Numercal results for structure-free confguraton (G) The reproducton of the expermental tests carred out n the absence of any structure has been performed by means of the standard HM solver and the seabed evoluton occurrng durng each test has been nvestgated. Hence the bottom profle of the central crosssecton of the doman has been plotted every 000s n order to analyze the gradual changes occurrng at the movable bed. Fgure 3 and Fgure 4 sho the seabed changes over the entre doman occurrng respectvely under ave forcng OS and OR. The horzontal plane s dentfed by tme evoluton (tme - axs) and spatal dstrbuton (x - axs) of the bottom profle. The vertcal dmenson represents the seabed depth. Blue refers to deeper aters and lght green to the emerged beach hle colors n beteen blue and green refer to shallo and ntermedate aters. Fgure 3 HM-solver results: cross-secton seabed evoluton n the absence of structures (confguraton G) under the rregular ave forcng OS. 56

76 Fgure 4 HM-solver results: cross-secton seabed evoluton n the absence of structures (confguraton G) under the regular ave forcng OR. The ntal bottom profle (at tme = 0 n both fgures) s gven by a pece-se slopng beach. The frst length (from x = 0 to x =.6m) s mldly slopng as ell as the second one (up to 8.5m) hle the thrd length (from 8.5m to 9.3m) s very steep almost a step change beteen offshore (deep/ntermedate aters) and nshore aters here the ater depth s d 0.m. Here the maxmum ave length calculated from the ave gauge measurements s L max m thus provdng d/l Hence numercal results comng from the HM solver may approprately approxmate the real seabed evoluton obtaned n the laboratory n the regon ncluded beteen x = 9m and x = 9m because nshore of the step change the shallo-ater approxmaton s satsfed. As a partal support to ths statement e observe that the most ntense morphologcal changes occur approxmately beteen the locaton 0m and the shorelne hch s dentfed n the fgures by a sold black lne. In fact after some tme steps the almost flat porton of the seabed (beteen 0m and 4m) s characterzed by some undulatons especally hen the ave forcng s regular (Fgure 4). Hoever the largest changes occur at the sash zone as ell as n the most nearshore zone. In ths zone the shorelne poston changes frequently (see especally Fgure 4) and a berm gradually forms out of the emerged beach. Such a berm stays stable hen the rregular (and less energetc) ave s forced (see Fgure 3) hereas t alternatvely rses and collapses more than once hen the regular ave s run (see Fgure 4 here the hte regon ust above the black shorelne thckens up to a maxmum then t decreases abruptly). The berm collapse may be due to the large steepness reached by the berm under the regular ave forcng. We beleve that though e used a bed-load formula hch 57

77 ncludes the beach slope stablzng effects a proper reproducton of the expermental condtons here some coheson surely characterzes the model sand cannot be acheved n ve of the overly smplfed transport formulatons Numercal results for the submerged-breakater confguraton (B) The reproducton of the expermental tests carred out n the presence of the submerged structure that s the closest to the shore has been performed by means of the modfed verson of the HM solver (see Appendx B) that accounts for a rgd body on the movable seabed. In the present confguraton the only ay for aves to reach the shore s by overpassng the structure. Thus the breakater cuts don the hghest aves thout any other flux contrbuton through the structure fltraton. The bottom profle of the central cross-secton of the doman has been plotted every 000s as for the prevous confguraton. Fgure 5 and Fgure 6 sho the seabed changes occurrng respectvely under the rregular ave OS and the regular ave OR. The ntal bottom profle (tme = 0) s characterzed by the same pece-se planar beach used for the structure-free confguraton (G) except for the breakater presence. Such a structure s placed n correspondence of the step-lke bathymetrc transton located beteen 8.5m and 9.3m. Hence the breakater separates n ths case the offshore from the nshore aters. Fgure 5 HM-solver results: cross-secton seabed evoluton n the presence of a submerged structure (confguraton B) under the rregular ave forcng OS. 58

78 Fgure 6 HM-solver results: cross-secton seabed evoluton n the presence of a submerged structure (confguraton B) under the regular ave forcng OR. In ths confguraton the most ntense morphologcal changes occur close to the shorelne dentfed n the fgures by the black lne although mportant seabed changes occur offshore of the breakater especally hen a regular ave s forced (Fgure 6). In fact the presence of the submerged breakater and the almost flat porton of the seabed located ust seaard of the very steep nner surf zone make the aves decrease thus not affectng sgnfcantly the seabed morphology. Large undulatons rse n correspondence of the more offshore regon hen the regular ave s run (Fgure 6). Actually the HM solver orks ell n shallo-ater condtons as explaned before hereas some errors may affect the treatment of the hydrodynamcs and consequently of the morphodynamcs occurrng n the outer surf zone (see.). Such seabed changes are essentally due to the reflecton provded by the breakater hch s larger n the case of the regular ave forcng than n the case of a spectrum (Fgure 5). Most of the bottom evoluton occurs n correspondence of the nearshore regon representng the steepest part of the beach. The generaton of an emerged bar close to the shorelne s as n the confguraton G an mportant feature of the bottom profle nduced by both spectral and regular aves. In the former case the bar remans stable (see Fgure 5) also because the JONSWAP spectrum s less energetc than the regular ave hch provdes after a thckenng of the hte regon over the shorelne an abrupt collapse at tme = 9000s of the berm (Fgure 6). Even n ths case the large steepness reached by the berm under a regular forcng provdes a stablty loss th respect to the real case. 59

79 Comparsons th expermental results (confguraton G) In the follong comparsons beteen expermental and numercal results obtaned for a structure-free beach are provded. Fgures llustrate the fnal seabed changes obtaned th both laboratory experments (n green) and numercal smulatons (sold red lnes) under a spectral (Fgure 7) and the regular (Fgure 8) ave forcng. The fnal ater level obtaned through the HM tests s n blue hle the ntal bottom profle s represented by a dashed red lne. The man features of both fgures are: ) the large sedment moblzaton n the steep regon beteen 4m and 4.5m and n the entre sash zone.e. up to 6m or 5m dependng on the ave type; ) the emerged bar generated shoreard of the stll-ater shorelne; 3)the good agreement beteen numercal and expermental results n the nner surf zone n partcular beteen x = m and x = 4m. Fgure 7 shos that numercal results approxmate farly ell the berm gron thn the sash zone n the expermental tests even f the HM solver provdes a smaller shorelne retreat the berm beng generated slghtly more offshore. The cause s the seaard sedment transport nduced by the HM solver that s reduced f compared th expermental results. Such a concluson seems to match th hat seen n Fgure of 4.3 here the suspended-load transport provded by the solver s manly onshore-drected after a sash event unlke the PH05 s analytcal result. Hoever n the flume tests many other varables are nvolved.e. to-dmensonal nature of the flo and further some expermental errors may lead to ether an mperfect morphodynamcs representaton or an ncorrect bathymetrc survey. Fgure 7 Seabed evoluton for confguraton G under the regular ave forcng OS: comparson of the fnal results of the HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. 60

80 Fgure 8 Seabed evoluton for confguraton G under the regular ave forcng OR: comparson of the fnal results of the HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. Fgure 8 shos a orse approxmaton of the expermental results th respect to that llustrated n Fgure 7. A far agreement can be seen n the regon (x = -4m) ncluded n the almost flat zone here some seabed oscllatons occur. The rse of a submerged bar n the expermental tests around x = 0m may have dsturbed the seabed changes n correspondence of the more nshore regon because t orked smlarly to the submerged breakater of confguraton B. Hence t provded a selecton of aves by cuttng out them and movng the ave breakng more nshore thus moblzng a very large amount of sedments n the vcnty of the shorelne. Such a behavor made the emerged berm ncrease largely. Moreover the submerged bar as ell as a breakater nduced a strong reflecton and generated a sgnfcant eroson/deposton pattern smlar to that observed n the expermental tests descrbed n Vcnanza et al. (00). Actually the submerged bar as asymmetrcal th respect to the flume axs and n Fgure 8 s represented through the mean beteen the seabed profles surveyed at the lateral sdes of the flume. The possble explanaton of such a laboratory effect s ether the nonhomogeneous compacton of the expermental sedment that as profled by hand after each test or the dfferent gran sze along the flume cross-secton. Fnally n the tests performed th the HM solver th aves forced over a structure-free beach the offshore porton of the beach (ater depth d -0..) s not ell smulated as expected because the model s assumptons are not approprate for reproducng the flo condtons of the outer surf zone. Inner surf zone and sash zone are nstead farly ell reproduced for both the flat regon (beteen 0m and 4m n the onshore drecton) and emerged berm. A orse berm smulaton s provded hen the regular ave s used. 6

81 Comparsons th expermental results (confguraton B) In the follong comparsons beteen expermental and numercal results obtaned n the presence of a submerged structure are provded. In Fgure 30 and Fgure 9 the sold red lne represents the HM numercal result hle the green lne s the fnal seabed profle found n the flume experments under respectvely the spectral ave OS and the regular ave forcng OR. The dashed red lne ndvduates the ntal bottom profle. As n the prevous fgures the man features are a large sedment moblzaton n the sash zone and an emerged bar shoreard of the stll-ater shorelne. In the nner surf zone (x = 0-4m) a good agreement beteen numercal and expermental results takes place. The numercal solver approxmates farly ell the berm evoluton found through the expermental test llustrated n Fgure 30 even f the HM solver provdes a smaller shorelne retreat. In fact the berm rose more offshore confrmng the same results obtaned for confguraton G (see Fgure 7). The offshore transport nduced by the HM solver s reduced f compared th expermental data as also seen n the comparson th the analytcal soluton provded by PH05 (see 4.3). Instead the protecton provded by the submerged structure makes the flat regon behnd the barrer reman almost stable except for a small scour at the toe of the structure tself that forms durng the experment and s farly ell reproduced by the HM solver. On the contrary sedments of the numercal seabed that are located offshore of the breakater are not sgnfcantly moblzed by aves hle the expermental seabed s largely modfed because of the reflecton nduced by the submerged barrer. Fgure 9 Seabed evoluton for confguraton B under the rregular ave forcng OS: comparson of fnal results comng from HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. 6

82 Fgure 30 Seabed evoluton for confguraton B under the regular ave forcng OR: comparson of fnal results comng from HM solver (sold red lne) and flume experments (sold green lne); ntal seabed (dotted red lne) and ater level at the fnal numercal output (n blue) are also shon. A orse reproducton of the expermental data s gven by Fgure 9. In the almost flat regon beteen x = 9.5m and x = 4m some oscllatons occur together th a lttle scour at the toe of the structure. The nstablty of the emerged berm s probably due to noncohesve nature of the sedments reproduced by the _HM solver hle the sand used n the flume as characterzed by a very lttle porton of cohesve grans. Further the bad reproducton of the flo n the outer surf zone.e. offshore of the breakater s due to the restrcted valdty feld of the NSWE are the man causes of the bad reproducton of the seabed changes. Laboratory effects also exst thus provdng expermental errors addng to the numercal ones. As an example the eroson/deposton pattern generated n the most offshore regon of the expermental doman s consstent th hat proposed by Sumer & Fredsøe (000) on the ave reflecton due to rubble-mound breakaters but t s larger than hat expected probably because of errors due to the sedment scalng. Hoever a large sedment moton s also evdent offshore of the barrer n the numercal smulatons even f the scale s dfferent th respect to the expermental results. Fnally n the tests performed th the HM solver n the presence of the submerged breakater the offshore sde of the beach s not ell smulated as expected because of the shallo-ater nature of the model. The nner surf zone and the sash zone are nstead qute ell reproduced for both flat regon located behnd the structure and emerged berm. A orse result s provded hen a regular ave s forced. 63

83 Chapter 5. Concludng Remarks In the present ork the development of a numercal model amed at reproducng both hydrodynamcs and morphodynamcs n the nearshore zone has been descrbed. Physcal modelng and feld experments have both been used to analyze the dynamcs at hand. One of the best models to descrbe the nearshore flos s based on the use of the NSWEs that approxmate farly ell the hydrodynamcs n the more onshore regon (nner surf zone and sash zone) but are less suted for the smulaton of the outer surf zone because of the overestmate of the ave breakng phenomenon. Startng from a robust avalable NSWE model a morphodynamc module has been developed based on the soluton of the Exner equaton. The constant update of both hydrodynamcs th the ne bathymetry and morphodynamcs th the velocty feld makes the code a eakly-coupled model. In fact equatons are not ntegrated at the same tme but solved separately and matched through a sequental splttng hch represents a good compromse beteen reduced computaton tme and qualty of the fnal soluton. For the morphodynamc treatment of the shorelne and for solvng the Exner equaton the same method employed for the hydrodynamc equatons has been used.e. the WAF method. It conssts n solvng a sub-grd Remann problem and n evaluatng the fnal flux at the ntercell pont as a eghted average of the fluxes occurrng at the closer cells. For the dscretzaton of the Exner equaton sutable closure las have been chosen to reproduce sedment transport. Because of the dfferent applcaton felds some formulas that are avalable n the lterature are very smple and depend only on a constant term that may vary over a rather de range. Other formulas depend on several terms and are dffcult to be ntegrated n any solver. Such a dffculty s at the bass of the larger dffcultes n mplementng a fully-coupled model hch makes use of realstcallycomplete and complex sedment transport closures. For ths reason the choce to ork th a eakly-coupled model has been made. It enables to choose among several transport formulas thout havng the constrant of usng alays the same la and does not need to compute a specfc characterstc ave structure. In fact t s supposed that the naccuraces assocated th the use of a eakly-coupled numercal solver are neglgble f compared th the uncertantes due to the choce of approprate sedment transport formulas. 64

84 To descrbe sedment fluxes three formulas have been chosen for the bed-load transport to for the suspended-load transport. The total transport rate s gven by the sum of both contrbutons. For the descrpton of the nearbed transport the smple las of Kelly (009) and Kelly and Dodd (00) have been chosen together th the more complcated relatonshp gven by Beso et al. (003). Suspended sedment transport has been modeled th the formula of Camenen & Larson (008) here the term gvng the reference concentraton has been descrbed through both formulaton of Camenen & Larson (008) and that of Van Rn (984). The valdaton of the numercal solver has been done by means of several tests. To test seres enabled to compare the performances of the bed-load transport descrbed through the to smplest formulas of Kelly (009) and Kelly and Dodd (00). The valdaton carred out by usng expermental and numercal tests descrbed n Fraccarollo & Capart (00) has revealed farly good performances of the model descrbng the front speed of a dam-break ave and the shape of the seabed evoluton hle the evoluton of the seabed especally n correspondence of the gate poston s not ell predcted. In fact the HM solver descrbed n the present thess as not desgned to specfcally reproduce the sheet flo transport analyzed by Fraccarollo & Capart (00). The second test seres amed at comparng the HM solver th the numercal/analytcal solutons of Kelly (009) has shon farly good performances of the HM model n the smulaton of ater depth velocty and seabed changes occurrng after a dam-break event. When the bed-load transport s represented as only dependent on the cubc velocty some spurous oscllatons characterze the seabed evoluton. Hoever f dependence on the ater depth s also ntroduced better trends thout oscllatons and n very good agreement th Kelly (009) are provded. The valdaton of the suspended transport has been done through the reproducton of the sash flos tested by Prtchard & Hogg (005) by means of an analytcal decoupled model. Qualtatvely the nstantaneous sedment fluxes found by Prtchard & Hogg (005) and through the HM solver (suspended transport descrbed by means of the closure la of Camenen & Larson 008) are farly smlar. Hoever three tests have been carred out th the numercal HM solver usng three dfferent frcton coeffcents: they hghlghted strong dfferences n the sedment transport rate.e. f a larger coeffcent s used a larger onshore transport occurs. Whereas Prtchard & Hogg (005) found a net offshore transport and clamed that all the decoupled models can only provde an offshore-drected sedment transport after a sngle sash event the soluton of the HM solver s a net transport that s every tme onshore-drected even varyng the frcton coeffcent. Other tests have been performed n order to check the adaptablty of the code at reproducng laboratory tests that have been carred out n confned condtons nsde the Hydraulcs Laboratory of the Unverstà Poltecnca delle Marche (Ancona). For ths purpose the bed-load transport has been descrbed by takng nto account the stablzng effect of gravty and the suspended-load transport through Camenen & Larson (008) s formulaton. To of the confguratons tested n the flume have been reproduced numercally: one thout any defense structures the other characterzed by a submerged barrer. Numercal results seem to agree partally th the seabed changes found n the 65

85 expermental tests. In fact hen spectral aves are run the man features of the fnal expermental profles are also present n the soluton found through the HM solver such as the emerged berm that rses closely to the shorelne or the small undulatons formng n the nner surf zone or the scour at the nshore toe of the submerged breakater. The predcted onshore sedment transport s larger n the numercal smulatons thus provdng a smaller berm retreat th respect to the expermental results. In the case of regular aves the seabed evoluton found th the HM solver s very poor for both confguratons only predctng farly good the bottom changes n the nner surf zone that s far enough from the shorelne. Then n the HM smulatons the emerged berm collapses after t reaches a certan heght probably because of the non-cohesve nature of the numercal sedments unlke the real granular materal used n the experments. A reason for the non-perfect matchng of expermental and numercal results s also the exstence of laboratory effects n the ave flume durng the expermental tests that provded asymmetrcal seabed evolutons along the cross-secton of the flume. At the end the comparson beteen numercal results of the HM solver and the expermental data s affected by dfferences that can be reduced through the smulaton of real ave condtons on a real sandy beach. Fnally the HM solver s a numercal solver able to predct offshore and onshore sedment transport scours at the toe of detached breakaters eroson/deposton patterns th a farly good approxmaton by means of a eakly-coupled approach that enables to choose among a seres of sedment transport formulas for the descrpton of both bed-load and suspended-load transport. 66

86 67

87 References Beso G. Blondeaux P. & Frsna P. A note on tdally generated sand aves J. Flud Mech. vol. 485 pp. 790 (003). Brocchn M. & Baldock T.E. Recent advances n modelng sash zone dynamcs: Influence of surf sash nteracton on nearshore hydrodynamcs and morphodynamcs Rev. Geophys. 46 (008). Brocchn M. Bernett R. Mancnell A. & Albertn G. An effcent solver for nearshore flos based on the WAF method Coast. Engng (00). Brocchn M. Drago M. & Ioventt L. The modellng of short aves n shallo aters. Comparson of numercal models based on Boussnesq and Serre equatons Int. Conf. Coastal Eng. ASCE 7688 (99). Brocchn M. & Peregrne D.H. Integral flo propertes of the sash zone and averagng J. Flud Mech (996). Buccno M. & Calabrese M. Conceptual approach for predcton of ave transmsson at lo-crested breakaters J. Wateray Port Coastal Ocean Eng. ASCE 33 (3) 3-4 (007). Calabrese M. Vcnanza D. & Buccno M. Verfcaton and recalbraton of an engneerng method for predctng D ave setup behnd submerged breakater Proc. Intern. Coastal Symp. 05 Hofn Iceland 005. Camenen B. & Larson M. A general formula for noncohesve suspended sedment transport J. Coast. Res (008). Camenen B. & Larroudé P. Comparson of sedment transport formulae for the coastal envronment Coast. Engng (003). De Vrend H.J. DH Mathematcal Modellng of Morphologcal Evolutons n Shallo Water Coast. Engng. -7 (987). De Vrend H.J. Zyserman J. Ncholson J. Roelvnk J.A. Péchon P. & Southgate H.N. Medum-term DH coastal area modellng Coast. Engng (993). Dng Y. Wang S.S.Y. & Ja Y. Development and Valdaton of a Quas-Three- Dmensonal Coastal Area Morphologcal Model J. Wateray Port Coastal Ocean Eng. ASCE 3 (6) (006). Fraccarollo L. & Capart H. Remann ave descrpton of erosonal dam-break flos J. Flud Mech (00). 68

88 Fredsøe J. Modellng of non-cohesve sedment transport processes n the marne envronment Coast. Engng (993). Grass A.J. Sedment transport by aves and currents SERC London Cent. Mar. Technol Report No. FL9 (98). Hamm L. Madsen P.A. & Peregrne D.H. Wave transformaton n the nearshore zone: a reve Coast. Engng (993). Harten A. Hgh resoluton schemes for hyperbolc conservaton las J. Comput. Phys (983). Kelly D.M. Bore-drven sash on a moble beach PhD thess School of Cvl Engneerng Unversty of Nottngham Nottngham UK (009). Kelly D.M. & Dodd D. Beach face evoluton n the sash zone J. Flud Mech (00). Longuet-Hggns M.S. Longshore currents generated by oblquely ncdent sea aves J. Geophys. Res (970). Lorenzon C. Mancnell A. Postacchn M. Mattol M. Soldn L. & Corvaro S. Expermental tests on sandy beach model protected by lo-crested structures Proc. 4 th Intern. Short Conf. on Appl. Coastal Res. Barcelona Span (009). Lorenzon C. Pattella A. Soldn L. Mancnell A. & Brocchn M. An expermental nvestgaton of the hydrodynamc crculaton n the presence of submerged breakaters Proc. of the 5 th Intern. Symp. on Ocean Measur. and Anal. (005). Masselnk G. & Puleo J.A. Sash-zone morphodynamcs Contn. Shelf Res (006). Madsen P.A. & Schäffer H.A. Hgher-order Boussnesq-type equatons for surface gravty aves: dervaton and analyss Proc. R. Soc. London A (998). Postacchn M. Lorenzon C. Mancnell A. & Brocchn M. Some expermental results on the effcency of dfferent breakater confguratons Att XXXII Conv. Naz. d Idraulca e Costruzon Idraulche Palermo Italy (00). Prtchard D. & Hogg A.J. On the transport of suspended sedment by a sash event on a plane beach Coast. Engng. 5-3 (005). Ruol P. Faedo A. & Pars A. Prove spermental sul comportamento d una scoglera a cresta bassa e sul fenomeno del plng-up a tergo d essa Stud Coster (003). Shapro R. Lnear flterng Mathematcs of Computaton (975). Shen M.C. & Meyer R.E. Clmb of a bore on a beach: part 3. Run-up Journal of Flud Mechancs (963). Stoker J.J. Water aves Interscence Ne York (957). Sumer B.M. & Fredsøe J. Expermental study of D scour and ts protecton at a rubblemound breakater Coast. Engng (000). 69

89 Toro E.F. Remann solvers and numercal methods for flud dynamcs Sprnger Berln (997). Van der Meer J.W. Conceptual desgn of rubble mound breakaters Proc. of the Short Course of the 3 rd Intern. Conf. on Coast. Engng. ASCE (99). Van Rn L. Sedment Transport part II: suspended load transport J. Hydr. Dv (984). Veeramony J. & Svendsen I.A. The flo n surf zone aves Coast. Engng (000). Vcnanza D. Contestable P. Postacchn M. & Brocchn M. Esperment n larga scala sull'effetto del raggruppamento delle onde sulla morfodnamca della zona d battga Att XXXII Conv. Naz. d Idraulca e Costruzon Idraulche Palermo Italy (00). Watson G. Barnes T.C.D. & Peregrne D.H. The generaton of lo frequency aves by a sngle ave group ncdent on a beach Proc. 4 th Intl. Conf. on Coast. Engng. - ASCE (994). Whtham G.B. Lnear and nonlnear aves John Wley & Sons Ne York (974). Zyserman J.A. & Johnson H.K. Modellng morphologcal processes n the vcnty of shore-parallel breakaters Coast. Engng. 45 (3) 6-84 (00). 70

90 Appendx A. The theoretcal approach n the laboratory experments The sash zone evoluton durng strong sea storms depends mostly on to dfferent mechansms for the submerged part and for the emerged part above the mean sea level. The eroson evoluton of the submerged part s due to the ave-nduced oscllatory and transportng currents the morphodynamcs of the emerged part depends manly on the runup or sash motons and for beaches th coarser sedments also on the permeablty. The hgh energy generated n the run-up oscllatory moton provdes rapd morphologcal changes assocated th the development of an emerged berm. These mechansms occur also hen detached breakaters are present. A typcal behavor s evdent n both submerged and emerged barrers durng a sea storm: an ncreasng of the stll ater level n the protected zone th respect to the seaard. Ths phenomenon s knon as plng-up and the mechansm that generates t s qute complex. It depends on a seres of terms provdng an equlbrum beteen the nshore and the offshore and sketched n Fgure 3. For an emerged breakater the plng-up s due to the ater passng over (Q ov ) and flterng through the structure (Q n ). Ths ncrease of the ater level n the nshore area leads to the search for an equlbrum of the mass fluxes hence the ater flters out through the structure (Q out ) and flos through the gaps located beteen to contguous breakaters generatng the rp currents (Q r ). In the case of submerged barrers the mechansm s almost the same though another outard component allong the ater to flo over the freeboard (Q fr ) s stll mportant. Several studes have been carred out n the last decades to understand analytcally the entre flux pattern for both submerged and emerged structures. A lot of tests have been carred out n D ave flumes hch produce the maxmum plng-up because of the laterally confned condtons hence the tested breakater can be seen as an nfntely long barrer n the real case. In the D expermental tests the emerged breakaters provde a plng-up n the protected zone and an outard flo through the structure that can be ether assocated th a recrculaton flo (Q r 0) representng the flo through the gaps or sngle (confned confguraton Q r =0). Ruol et al. (003) carred out a lot of set-up tests and nvestgated several condtons by runnng dfferent ave types and by applyng dfferent recrculaton condtons startng from the confned condton (Q r =0) tll the maxmum recrculaton assocated th a null plng-up (δ h =0). The latter case refers to nfntely de gaps the confned condton to nfntely long breakaters. A lnear dependence as found beteen 7

91 plng-up (δ h ) and the net transmtted dscharge (Q r =Q ov Q n -Q out ) dependng on the ave type. ave Qov Qn δh plng-up m..l. Qout Qr ave Qov plng-up Qfr δh m..l. Qout Qn Qr Fgure 3 Sketch of the flos nterestng the plng-up phenomenon n the presence of emerged (top fgure) or submerged (bottom fgure) breakaters. Accordng to the CVB method (see Calabrese et al. 005) a balance equaton governs the hydrodynamc equlbrum around a submerged breakater: here S xx Π P = 0 (A.) S xx s the radaton stress gradent for the xx component Π the mean force of P the hydrostatc thrust dfference. The resoluton the structure on the ater volume and of (A.) yelds to: m [ b ( b 4 )] 0. 5 δ (A.) = 0.5 c that s the plng-up due to the momentum flux equlbrum. Besdes the mass balance produce another plng-up contrbuton: q δ q = B eq (A.3) f R 0 3 c here f s the frcton factor and q the flo rate. Terms b c and B eq n equatons (A.) and (A.3) depend on several parameters: poston of the breakng pont (x b ) breakng depth (d b ) breakater freeboard (R c ) ncdent sgnfcant ave heght (H m0 ) transmsson coeffcent (K t ) breakater geometry. 7

92 Appendx B. Rgd bodes n the doman The possblty to have rgd bodes n the doman s necessary to examne erosons and accretons around submerged and emerged coastal structures. For ths scope a rgd-layer has been added to the doman of the HM solver. Ths knd of update can be easly done by means of the fnte-volume dscretzaton. In fact cells are taken as free from sedments and for ths purpose the outgong fluxes are blocked. Hence n the HM solver three dfferent layers are defned: ) the sea level ) the surface of the movable seabed and 3) the surface of the rgd bottom. The ater level s never under the movng seabed and the latter s never under the rgd layer thus provdng a representatve sketch lke that proposed n Fgure 3. When ater s absent and the beach s dry sea level and movng seabed are consdered as perfectly matchng hle f no movng materal s present on the beach surface.e. f the rgd body s assumed at a hgher level than the movable beach both seabed layers match. Fgure 3 Representatve sketch of the three-layer dstrbuton: sea level movng seabed and rgd seabed. In the case here rgd and movng layers match the sedment s enabled to move only to accrete and not to erode. In fact hen a rgd body s present on the beach surface t can enable erosonal patterns occurrng at the lateral boundares but scours are not alloed on t. At the same tme depostons on the body surface are consstent th real cases. Wthout the rgd layer the elevaton of the movng seabed s computed through the formula: 73

93 n ( / n t / n ) ( / n H H J J / ) n n t z = z / / / / (B.) x y here H and J are respectvely fluxes evaluated along x and y (see Fgure 33) hle z s the movng bottom depth (see for more detals.3). n/ J / n/ H -/ n/ J -/ n/ H / Fgure 33 Sketch of fluxes enterng and escapng the cell ( ) at tme n/. In the presence of the rgd layer placed at a depth zr th respect to the stll ater depth n n z zr fluxes have been changed by means of a smple procedure. If t means that there s no sand because both seabed layers match. In ths case ncomng fluxes are alloed hereas outgong fluxes are neglected and set equal to zero.e. H H n / / n / / On the other sde f = max = mn n ( / n 0) / n H max( / / J / = J / 0 ) n ( / n 0) / n H J = mn( J / 0) / n n z zr Hoever f at the same tme / / (B.) > there s enough sand to be ether deposted or removed. n n z < zr t means that movng seabed and rgd layer do match after tme step n because all the sand has been eroded beteen n and n. In ths case the outgong fluxes have to be reduced n order to make the cell free from sedments at tme step n hence: H H n / / n / / = max = mn n ( / n / n ) / n max( / n H / / αh / J / = J / αj / ) n ( / n / n ) / n mn( / n H αh J = J αj / ) / / / here α s the reducton coeffcent defned as: / / (B.3) 74

94 z zr n n = n n z z α. (B.4) 75