General longshore transport model

Coastal Engineering 71 (2013) 28–36

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General longshore transport model Giuseppe R. Tomasicchio a,⁎, Felice D'Alessandro a, Giuseppe Barbaro b, Giovanni Malara b a b

University of Salento, Engineering Dept., Via per Monteroni, Ecotekne, 73100 Lecce, Italy Mediterranea University of Reggio Calabria, MecMat Dept., via Graziella 1, 89060 Reggio Calabria, Italy

a r t i c l e

i n f o

Article history: Received 7 May 2012 Received in revised form 27 July 2012 Accepted 29 July 2012 Available online 30 August 2012 Keywords: Longshore transport Reshaping or berm breakwaters Gravel beaches Sandy beaches Mobility index

a b s t r a c t In the present paper a general longshore transport (LT) model is proposed after a re-calibration of the model originally introduced by Lamberti and Tomasicchio (1997) based on a modified stability number, N⁎⁎ s , for stone mobility at reshaping or berm breakwaters. N⁎⁎ s resembles the traditional stability number (Ahrens, 1987; van der Meer, 1988) taking into account the effects of a non-Rayleighian wave height distribution at shallow water (Klopman and Stive, 1989), wave steepness, wave obliquity, and nominal diameter of the units. Nine high-quality data sets from field and laboratory experiments have been considered to extend the validity of the original model for a wider mobility range of the units: from stones to sands. The predictive capability of the proposed model has been verified against the most popular formulae in literature for the LT estimation of not cohesive units at a coastal body. The comparison showed that the model gives a better agreement with the physical data with respect to the other investigated formulae. The proposed transport model presents a main advantage with respect to other formulae: it can represent an engineering tool suitable for a large range of conditions, from sandy beaches till reshaping breakwaters. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Typically a coastal structure is intended as a body which reduces and absorbs the wave energy. In this sense, both a rubble mound breakwater and a beach are bodies made of not cohesive units functioning as a coastal structure. Different coastal structures can be identified by means of the stability number, Ns = Hs/ΔDn50, where Hs = significant wave height at the toe of the structure, Δ = relative mass density and Dn50 = nominal diameter of the units composing the structure (van der Meer, 1988). In particular, structures such as caissons or structures with large armour units are characterized by small values of Ns; large values imply gravel beaches and sandy beaches. In this path, berm breakwaters (Baird and Hall, 1984; Hall et al., 1983; Kao and Hall, 1990) are designed with a rather steep seaward slope and a horizontal berm just above the still water level: the first storm develops a more gentle profile which does not change later on (van der Meer, 1988); resembling the behaviour of a beach, berm breakwaters profile changes are expected to be important and the stability of the coastal body is based on the concept of dynamic stability. This concept was introduced by van der Meer (1988, 1992): units are displaced by the wave action until a profile is reached where the transport capacity along the profile is reduced to a minimum for a given wave condition. Material around the still water level is continuously moving during each run-up and run-down of the waves. An influence from the rock shape on the reshaping process has been ⁎ Corresponding author. E-mail address: [email protected] (G.R. Tomasicchio). 0378-3839/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coastaleng.2012.07.004

investigated by several authors (e.g. Frigaard et al., 1996). An overview of types of structures for different Ns values is given in Table 1 (van der Meer, 1992). Dynamic stability can roughly be classified by Ns > 4 (van der Meer, 1988, 1992). Determination of the reshaped profile developed under the dynamic stability concept requires a constant crosssectional area (zero net cross-shore transport) during the wave attack; this means that, when discussing any dynamically stable coastal body, an estimate of the amount of units moved as a “longshore transport” (LT) along the longitudinal axis of the structure should be taken into account. As a consequence, the knowledge of the LT and its cross-shore distribution pattern in the surf zone are central in coastal engineering studies; practical engineering applications, such as the design of dynamically stable reshaping or berm breakwaters, dispersion of the beach-fill and placed dredged material, beach nourishment projects, sedimentation rates in navigation channels, they all require accurate predictions of the LT. A possible classification of the LT methods distinguishes three types. The first type comprises the energetic methods and can be divided into two sub-types: the energy flux approach and the stream power approach. The energy flux approach (Bayram et al., 2007; Inman and Bagnold, 1963; Komar, 1969; Komar and Inman, 1970; USACE, 1984) was developed specifically for coastal sediment transport, while the stream power approach (Bagnold, 1963, 1966; Bailard, 1981) is of more general application to any sediment transport situation, including fluvial environment. The second type is based on the force-balance method (Bijker, 1967, 1992; Damgaard

G.R. Tomasicchio et al. / Coastal Engineering 71 (2013) 28–36 Table 1 Classification of types of structures for different Ns values (van der Meer, 1988, 1992). Ns

Type of structure

1–4 3–6 6–20 15–500 >500

Statically stable breakwaters S-shaped and berm breakwaters Rock slopes/beaches Gravel beaches Sandy beaches

and Soulsby, 1996; Frijlink, 1952) which, classically, is the alternative method to the stream power approach in predicting sediment transport (van Wellen et al., 2000). The third type comprises formulae which were derived by a dimensional analysis method with parameters which are thought to be of importance for LT (Kamphuis, 1991; Schoonees and Theron, 1993; van der Meer, 1990; van Hijum and Pilarczyk, 1982). Table 2 presents a classification of the LT methods. More details on the three methods can be found in van Wellen et al. (2000). It should be noticed that the most popular LT formulae (e.g. Kamphuis, 1991; USACE, 1984) were developed predominantly for sandy, dissipative beach environments and have rarely been validated at beaches of coarser grain size or shingle beaches where a steeper slope results in a balance of hydrodynamic processes which is often very different from that at sandy beaches (van Wellen et al., 2000). In fact, a shingle beach is composed of gravel or small- to medium-sized cobbles: the sorting ranges from 2 to 200 mm diameter. Table 3 gives details on the specific ranges of the grain sizes of the commonly granular materials composing a beach. For the case of shingle beaches, LT is determined by the bed load and not by a combination of bed load and suspended load as in the case of a sandy beach. LT at gravel beaches is influenced by a steeper beach slope, typically 1:8, which encourages waves to form rapidly plunging or surging breakers close to the shoreline; thus, most of the energy dissipation is restricted to a narrow region that includes the swash zone (van Wellen et al., 2000). A well sorted coarse sediments mound also exhibits a larger porosity compared to the sand; this allows infiltration of water during the swash run-up, which weakens the backwash and can be identified with the formation of the berm at the run-up maximum (van Wellen et al., 2000). These phenomena lead to a different mode of energy dissipation compared to sandy beaches, which may partially invalidate the most popular formulae (e.g. Kamphuis, 1991; USACE, 1984) for LT estimates. Therefore, research on LT at gravel and shingle beaches has been performed in the past to deal with the erosion problems along these types of beaches, which are quite common along mid- and high latitude (formerly glaciated) parts of the world (van Rijn, 2002). As reported by van Wellen et al. (2000), only a few formulae (Chadwick, 1989; van der Meer, 1990; van der Meer and Veldman, 1992; van Hijum and Pilarczyk, 1982) have been derived specifically for LT at gravel beaches and most of them have been calibrated for a very limited data sets.

Table 2 Classification of the LT methods. LT methods Energetic

References Energy flux approach Stream power approach

Force-balance Dimensional analysis

(CERC equation, USACE, 1984); Inman and Bagnold (1963); Komar (1969); Komar and Inman (1970); Bayram et al. (2007) Bagnold (1963, 1966); Bailard (1981) Frijlink (1952); Bijker (1967, 1992); Damgaard and Soulsby (1996) van Hijum and Pilarczyk (1982); van der Meer (1990); Kamphuis (1991); Schoonees and Theron (1993)

29

Table 3 Basic classification of sediments (after Wentworth, 1922). Size range [mm]

Sediment name

0.05–2 2–64 64–256

Sand Gravel Cobble

{shingle

For the case of reshaping or berm breakwaters, Burcharth and Frigaard (1987) performed model tests to establish the incipient conditions for LT. Model tests corresponded to Ns = 3.5–7.1, with angles of wave attack of 15 and 30 degrees. Van der Meer and Veldman (1992) performed model tests with a berm breakwater with obliquity equal to 25 and 50 degrees and proposed a formula to estimate the LT at berm breakwaters and beaches made of coarse gravel. More details on the commonly adopted formulae for LT estimation for the various types of coastal structures can be found in section 2 of the present manuscript. Although the computation of reliable estimates of LT remains of considerable importance in coastal engineering design practice, at present the understanding of the LT phenomena is still rudimentary, being mostly based on empirical assumptions; the use of the different existing LT formulae only provides order-of‐magnitude accuracy between computed and observed LT rates. On the contrary, the increasing demand for accurate design wave conditions and input data for the investigation of sediment transport and surf zone circulation has resulted in a significant improvement of wave transformation models during the last decades (BTE models: e.g. Nwogu, 1993; Wei et al., 1995; Mild slope equation: e.g. Liu, 1990; RANS models: e.g. Lin and Liu, 1998a,b; Losada et al., 2000) allowing today to analyse several dynamic processes in wave and structure or beach interactions, such as wave dissipation due to breaking (D'Alessandro and Tomasicchio, 2008; Kennedy et al., 2000; Zelt, 1991) and run-up (Fuhrman and Madsen, 2008), which were only possible by means of physical models in the past. Because of the limited range of validity in the existing models, a valuable goal in the LT modelling is to make available a simple but ‘general’ model valid for a wider interval of units composing the structure, from stones to sands. The new procedure results from a re-calibration of the model originally derived by Lamberti and Tomasicchio (1997) for stones mobility at reshaping or berm breakwaters. The re-calibration of the original model (Lamberti and Tomasicchio, 1997) is based on an extensive database covering a wide range of wave and structure conditions. When identifying data sets suitable for the re-calibration procedure, accuracy and reliability in the measurements were primary criteria for the inclusion. In particular, nine high-quality data sets on hydrodynamics and sediment transport, collected during both field and laboratory experiments, have been adopted to calibrate and verify the new general LT model. The data obtained from a series of field campaigns at the U.S. Army Corps of Engineers Field Research Facility at Duck, NC, including the Duck 85 (Kraus et al., 1989), and SandyDuck (Miller, 1998, 1999) have been considered together with the data sets acquired by Wang et al. (1998) and Schoonees and Theron (1993). With regard to laboratory experiments, the data collected at the Large-Scale Sediment Transport Facility (LSTF) of the Coastal and Hydraulics Laboratory of the US Army Engineer Research and Development Center in Vicksburgh, MS, (Smith et al., 2003) have been adopted. In addition, the database comprises the data set from van Hijum and Pilarczyk (1982) for the case of gravel beaches, and the data sets from Burcharth and Frigaard (1987, 1988); van der Meer and Veldman (1992) and from the model tests carried out at the Danish Hydraulics Institute (Alikhani et al., 1996; DHI, 1995) as part of the EU-funded MAST2 programme for the estimation of the LT at reshaping rubble mound breakwaters.

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The paper reviews the existing LT formulae focusing on their range of validity, from berm breakwaters to sandy beaches. The new general model is then presented. Afterwards, the nine data sets adopted to calibrate and verify the proposed LT model are discussed. Finally, the suitability of the model is assessed by means of the comparison between the LT predictions and measurements from the entire adopted data base.

using high-quality field and laboratory data or through some theoretical considerations. Van Hijum and Pilarczyk (1982) proposed the following formula specifically valid at gravel beaches; Eq. (4) has been obtained from the laboratory experiments conducted by van Hijum (1976) and van Hijum and Pilarczyk (1982) at Delft Hydraulics: 1=2

Q LT −4 H s;d ð cos θÞ ¼ 7:1210 D90 gD290 T s

2. Commonly adopted formulae for LT estimation The most popular formula for LT estimation is commonly known as the CERC equation (USACE, 1984). This formula is based on the principle that the LT rate, without a distinction between bed load and suspended load, is proportional to the longshore wave power, P: Q LT ¼ K⋅P ¼

pffiffiffiffiffiffiffiffiffiffiffi ρK g=γ b 2:5 H sinð2θb Þ 16ðρs −ρÞð1−aÞ s;b

ð1Þ

where QLT is the transport rate in volume per unit time, K is an empirical coefficient, ρ is the density of water, ρs is the density of sand, g is the acceleration due to gravity, a is the porosity index (≅ 0.4), Hs,b is the significant wave height at breaking, γb is the breaker index (= Hsb/hb, with hb = water depth at breaking), and θb is the wave angle at breaking. There is no direct inclusion of the influence of the grain size in Eq. (1), other than via the coefficient K, which has been found to be quite variable even at sandy beaches (Dean, 1987; USACE, 2001). The study conducted by van Wellen et al. (2000) confirmed the reduction in K for coarse-grained sediment and indicated that the value of K when in the presence of gravel beaches is around 30% of the value of K from a sandy beach, although other values of K from field experiments of coarse-grained beaches have been much lower; for instance, Nicholls and Wright (1991) found K to be between 1% and 15% of that for sand, whereas Chadwick's (1989) trap data suggested a K value 7% of that for sand. This adds further inaccuracy about the LT estimation at gravel/cobbles beaches when using the CERC equation which remains site specific. The effects of the wave period and beach slope, which both influence wave breaking, and the grain size neglected in the CERC formula, have been considered by Kamphuis (1991), resulting in a more refined equation. The formula which Kamphuis (1991) found to be applicable to both field and laboratory data at sandy beaches is: 2

1:5

0:75

Q LT;m ¼ 2:27H s;b T p mb

−0:25

Dn50

0:6

sin

ð2θb Þ

ð2Þ

in which QLT,m is the transport rate of the immersed mass per unit time, and mb is the beach slope at breaking. Kamphuis also investigated whether Eq. (2) was applicable to gravel beaches by comparing its predictions with the experimental results of Van Hijum and Pilarczyk (1982). He found that it over-predicts these results by a factor of 2 to 5, concluding that this was to be expected, since gravel beaches absorb substantial wave energy by percolation. More recently, Bayram et al. (2007) proposed a predictive formula for the LT rate derived from both field and laboratory data at sandy beaches; it was developed from principles of sediment transport physics assuming that breaking waves mobilize the sediment, which is subsequently moved by a mean current: Q LT ¼

ε F V ðρs −ρÞð1−aÞgws

ð3Þ

"

1=2

Hs;d ð cosθÞ D90

# −8:3

sin θ   tanh 2πh L

ð4Þ

where Hs,d is the local significant wave height, Ts is the significant wave period, L is the wave length, h is the water depth, θ is the wave angle and D90 is the 90% representative grain diameter. According to van Wellen et al. (2000), Eq. (4) introduces unwanted complications by using wave parameters measured at an offshore location and at the toe of the structure. Chadwick (1989) recasted the Delft experimental data in terms of conditions at the breaking position to give:   2 Q LT ¼ 0:0013 gD90 T s W ðW−8:3Þ sin θb

ð5Þ

where: W¼

H s;b

pffiffiffiffiffiffiffiffiffiffiffiffi cos θ D90

ð6Þ

Van der Meer (1990) also re-analysed the van Hijum and Pilarczyk (1982) formula in order to make it more practical: Q LT ¼ 0:0012gDn50 T p H s

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi H s cos θb

! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos θb −11 sin θb ½kg=s Dn50

ð7Þ

Van der Meer and Veldman (1992) specified that Eq. (7) should only be applied within the limit Ns = 12–27, i.e. fairly large gravel in prototype. Eq. (7) shows a dependency on the grain diameter. For small grain sizes (gravel/sandy beaches) Eq. (7) reduces to: Q LT ¼ 0:0012πH s c sin 2θb

ð8Þ

where c = wave celerity. It is noticed that in Eq. (8) the diameter or grain size is not present in accordance with the CERC formula. Burcharth and Frigaard (1987, 1988) performed model tests to establish the incipient LT conditions for reshaping or berm breakwaters. LT is not desirable at berm breakwaters and therefore Burcharth and Frigaard (1987, 1988) gave the following recommendations for the design of berm breakwaters, which in fact correspond to the incipient conditions: - for trunks exposed to steep oblique waves - for trunks exposed to long oblique waves - for roundheads

Ns b 4.5; Ns b 3.5; Ns b 3.0.

Vrijling et al. (1991) adopted a probabilistic approach to calculate the LT at a berm breakwater over its total lifetime. In that case the start or onset of LT is extremely important. They used the data of the tests performed by van der Meer and Veldman (1992) and the data by Burcharth and Frigaard (1987), but not the extended series described in Burcharth and Frigaard (1988). Based on all data points they come to a formula for berm breakwaters: 

where F (= Ecg) represents the flux of wave energy towards shore, with E = wave energy and cg = wave group celerity, V is the mean longshore current velocity over the surf zone, ε is the transport coefficient which expresses the efficiency of the waves in keeping sand grains in suspension and ws is the fall speed; it may be determined

Q LT

Q LT ¼ 0 f or N s T p b100  2 ¼ 0:000048 N s T p −105 ½units=wave

ð9Þ

wave period related to the nominal where Tp⁎ is the dimensionless pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diameter: T p ¼ T p g=Dn50 :

G.R. Tomasicchio et al. / Coastal Engineering 71 (2013) 28–36

Eq. (9) slightly overestimates the start of LT; therefore Eq. (9) was adjusted to better determine the incipient LT: 

Q LT

Q LT ¼ 0 for Ns T p <105  2  ¼ 0:00005 Ns T p −105 ½units=wave

ð10Þ

Eq. (10) yields for wave obliquities roughly between 15 and 35 degrees. For smaller or larger wave angles the LT will substantially be less.

31

In the case of head-on wave attacks, under the assumption that, offshore the breaking point, the wave energy is negligible and that waves break as shallow water waves the following relations holds: 2 2

2 2

F ¼ 1=8ρgH0 cg;0 ¼ 1=8ρgHb cg;b

ð13Þ

where cg,0 is the offshore wave group celerity, cg,b is the wave group celerity at breaking, H0 is the offshore wave height and Hbpis the ffiffiffiffiffiffiffiffiffiffi wave height at breaking. Considering Eq. (13) and cg;0 ¼ 1=2 g=k0 ; where k0 is the offshore wave number, the longshore component of F can be written as:

3. Development of a general model for LT 5=2 −1=2

A ‘general’ model is defined relating LT due to oblique wave attacks to the mobility level of the units composing the coastal body (Lamberti and Tomasicchio, 1997). The LT model is based on the assumption that movements statistics is affected by obliquity only through an appropriate mobility index and that the units move during up- and down-rush with the same obliquity of breaking and reflected waves at the breaker depth (Lamberti and Tomasicchio, 1997; Tomasicchio et al., 1994). A particle will pass through a certain control section in a small time interval Δt if and only if it is removed from an updrift area of extension equal to the longitudinal component of the displacement length, ld sin θd, where ld is the displacement length and θd is its obliquity (Fig. 1). Assuming that the displacement obliquity is equal to the characteristic wave obliquity at breaking (θd = θk,b), and that a number Nod of particles removed from a Dn50 wide strip moves under the action of 1000 waves, then the number of units passing a given control section in one wave is: SN ¼

   ld N ⋅ od sin θk;b ¼ f N s ½units=wave Dn50 1000

ð11Þ

F cos θ∝H 0 s0



Ns ¼

Hk C k ΔDn50

sm;0 sm;k



2=5

γb 4k0 H 0

Hb ¼ H0

1=5

−1=5

¼ pH0 s0

ð15Þ

Komar and Gaughan (1972) found the best agreement with field and laboratory data assuming that γb = 1.42 or the proportionality constant p = 0.56. It follows that, considering the characteristic wave height at breaking, Hk,b, and sm,0 = sm,k = 0.03, Ns**can be also written as: 

Ns ≅

0:89Hk;b C k ΔDn50

ð16Þ

and it can be noticed that, according to the proposed LT model, the relevant parameter is the onshore energy flux and that the proposed model belongs to the category based on an energy flux approach. According to the refraction theory for plane and monotonically decreasing profiles, Hk,b and sin θk,b, can be evaluated as in the following (Lamberti and Tomasicchio, 1997; Tomasicchio et al., 1994): H k;b ¼

ð cos θ0 Þ

ð14Þ

and cos θ is present in Eq. (12) with a power 2/5. Eq. (13) related to γb = Hb/hb implies:

where: !−1=5

cos θ

 qffiffiffiffiffiffiffiffiffiffiffi2=5 2 H k cg cos θ γ b =g

ð17Þ

ð12Þ sin θk;b ¼

is the modified stability number (Lamberti and Tomasicchio, 1997) with: Hk = characteristic wave height; Ck = Hk/Hs; θ0 = offshore wave obliquity; sm,0 = mean wave steepness at offshore conditions and sm,k = characteristic mean wave steepness (assumed equal to 0.03). Lamberti and Tomasicchio (1997), reported that Hk is to be considered equal to H1/50, but H2% can also be adopted. In the first case: Ck = 1.55, in the second case: Ck = 1.40. The second factor in Eq. (12) is such that Ns* * ≅ Ns for θ0 = 0 if sm,0 = sm,k. For a berm breakwater, strict threshold conditions correspond to Ns* * ≅ 2.

ck;b ¼

ck;b sin θ c

ð18Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gHk;b =γb

ð19Þ

where ck,b is the characteristic wave celerity at breaking depth. The displacement length is calculated as (Lamberti and Tomasicchio, 1997): ld ¼



ð1:4Ns −1:3Þ Dn50 tanh2 ðkhÞ

ð20Þ

with k = wave number. Nod has been determined following a calibration procedure based on the least-squares method taking into account the full database. In particular, two different approximating functions are considered; to accommodate the calibration procedure, Nod values calculated from measured data are partitioned into two subintervals. The first interval refers to N⁎⁎ s ≤23: from berm breakwaters to gravel beaches. The second one re⁎⁎ lates to N⁎⁎ s >23: the interval for sandy beaches. For Ns ≤23, a third order polynomial approximating function provides a satisfactory agreement as shown by Tomasicchio el al. (2007). For N⁎⁎ s >23 a good agreement is given by a linear regression in log–log plane. After the adopted calibration procedure Nod is given as: Fig. 1. Definition sketch for the ‘general’ LT model.

Nod ¼

 2 20N  s ðN s −2Þ  exp½2:72 lnðNs Þ þ 1:12

N s ≤23  Ns > 23

ð21Þ

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G.R. Tomasicchio et al. / Coastal Engineering 71 (2013) 28–36

The estimated correlation coefficient results equal to 0.89 for Ns* * ≤ 23, and 0.92 for Ns* * > 23, respectively. LT rate can be also expressed in terms of [m 3/s] as in the following: Q LT ¼

SN D3n50 Tm

ð22Þ

sand, gravel and shingle. LT data were obtained using three different methods, namely tracers, samplers and traps, and measured morphological volume change. They pointed out a lack of information for grain sizes coarser than 0.6 mm. The principal reason for the lack of data is due to the difficulties encountered to measure LT when at gravel and shingle beaches. A deeper description of these aspects is given in van Wellen et al. (2000).

with Tm = mean wave period. 4. LT data Nine high-quality data sets of LT from field and laboratory experiments have been considered to calibrate and verify the proposed model for a wide mobility range of units: from stones to sand. In total, the nine data sets consist of 245 cases. Most of these cases (69 %) concern LT measurements at sandy beaches, whereas only 4.5% of the tests from the data set by van Hijum and Pilarczyk (1982) have been performed at gravel beaches; the rest of data (26.5 %) is referred to LT at berm breakwaters. Table 4 summarizes the characteristics of the considered data sets; it lists the different data sets with increasing Hs,b/ΔDn50 and includes the hydrodynamic conditions, sediment properties and observed LT rates, QLT,o. It is noticed that the wave angle at breaking was not measured during the Duck 85 experiment (Kraus et al., 1989). A short description of the different experimental field and laboratory methods adopted to determine the LT rates for the considered test cases is provided in the following sections. 4.1. Field data

Wang et al. (1998) Wang et al. (1998) performed field experiments along the Southeast coast of the United States and the Gulf coast of Florida between September 1993 and May 1995. They measured the total rate of LT in the surf zone, mostly by using streamer traps similar to the ones used in the Duck 85 study (Kraus et al., 1989). The streamer cloth was 2 m long and its mouth was 0.15 m wide and 0.09 m high. The mesh size of the sieve cloth was 63 μm. SandyDuck (Miller, 1999) During the SandyDuck 85 field data collection project, also performed at FRF, direct measurements of velocities and suspended sediment concentrations were conducted at a minimum of nine positions across a barred profile to provide an estimate of the total suspended LT (Miller, 1999). The suspended LT at each position was computed by multiplying the instantaneous concentration with the instantaneous longshore current velocity and time averaging the products. Wave measurements are available from a directional pressure gauge array located in a depth of 8 m offshore. The velocity and concentration measurements were taken by means of instruments attached to the lower boom of a track-mounted crane (Sensor Insertion System; SIS) stationed on the pier. The SIS can place instrumentation on the bottom in depths of up to 9 m and 15 m away from the pier pilings. The bed load LT was not measured.

Duck 85 (Kraus et al., 1989) The SandyDuck field data collection project was performed at the Field Research Facility (FRF) of the U.S. Army Corps of Engineers, located along the Atlantic coast near the village of Duck, North Carolina, USA (Kraus et al., 1989). LT was measured using portable streamer traps in the surf zone (depth is about 1 m). The traps consisted of flexible polyester filter cloth (0.105 mm mesh) which allows water to pass through but retains sediment of the nominal diameter larger than the mesh size. Each streamer cloth was 2 m long and its mouth was 0.15 m wide and 0.09 m high. Six or seven traps were simultaneously deployed along a line crossing the surf zone with the streamer mouths facing into the longshore current (sampling time of about 5 to 10 min).

Van Hijum and Pilarczyk (1982) A limited number of 3D experiments on gravel sized material have been carried out by van Hijum and Pilarczyk (1982) at the ‘de Voorst Laboratory’ of the Delft Hydraulics. LT was measured from beach profile surveys using the principle of continuity of sediment in the longshore direction. From these tests van Hijum and Pilarczyk (1982) derived the analytical equation which subsequently became known as the Delft equation as reported in section 2 (Eq. (4)).

Schoonees and Theron (1993) Schoonees and Theron (1993) made an inventory of the available field data suitable for evaluating LT rates, resulting in 123 data sets. The data were collected at beaches from a variety of sites all around the world, encompassing particulate (not-cohesive) sediment including

Burcharth and Frigaard (1988) Burcharth and Frigaard (1988) investigated the stability of reshaping rubble mound breakwaters in a 3D physical model experiment at the Hydraulics Laboratory of the Aalborg University. The sea states were chosen in the range from very little erosion to fast erosion

4.2. Laboratory data

Table 4 LT database. Data source

Hs,b [m]

Tp [s]

θb [deg]

Dn50 [mm]

USDA⁎

Hs,b/ΔDn50 Structure

Field–lab

QLT,o [m3/s]

DHI (1995)

0.07–0.15

1.2–2.2

−30–45

23

2–4 BB⁎⁎

L

3.1∙10−8–6.6∙10−6

van der Meer and Veldman (1992)

0.08–0.19

1–2

25–50

26

3.0–3.9 BB⁎⁎

L

9.3∙10−4–1.1∙10−2

Burcharth and Frigaard (1987, 1988)

0.08–0.19

1.5–2.5

15–30

16.9

3.5–7.1 BB⁎⁎

L

2.4∙10−7–1.9∙10−5

van Hijum and Pilarczyk (1982) Wang et al. (1998) Schoonees and Theron (1993) LSTF (Smith et al., 2003) Duck 85 (Kraus et al., 1989) SandyDuck (Miller, 1999)

0.1–0.2 0.2–1.1 0.2–3.4 0.2–0.3 0.8–1.2 1.7–4.3

1.0–2.5 3.0–10.5 5.0–12.0 1.5–3.0 9.0–13.1 6.0–13.0

30 2–20 2–35 6–7 [−] 5–23

4.0 0.2–2.25 0.18–1.0 0.15 0.17 0.17

Gravel⁎ - model Stone prototype Gravel⁎ - model Stone prototype Gravel⁎ - model Stone prototype Gravel⁎ Fine sand - gravel⁎ Fine - medium sand⁎ Fine sand⁎ Fine sand⁎ Fine sand⁎

28–57 GB⁎⁎ 145–13.5∙103 SB – GB⁎⁎ 245–55∙103 SB - GB⁎⁎ 3.2∙103–4.9∙103 SB⁎⁎ 11∙103–17∙103 SB⁎⁎ 24∙103–62∙103 SB⁎⁎

L F F L F F

9.9∙10−6–1.9∙10−4 6.4∙10−5–4.7∙10−3 8.6∙10−4 - 0.14 4.0∙10−5–1.1∙10−4 2.8∙10−4–1.9∙10−3 9.6∙10−2–0.6

⁎ ⁎⁎

USDA soil taxonomy (http://soils.usda.gov). BB = berm breakwater, GB = gravel beach, SB = sandy beach.

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of the profile under oblique wave attacks. The LT was found from video recordings of the movements of coloured stones placed in three bands over the profile. Moreover, after a specific number of waves (or time) the number, the positions and the total weight of each type of coloured stones were recorded. The band width and the number of waves were adjusted to the sea state in such a way that within the test period the non-coloured stones upstream the coloured bands did not pass the downstream coloured band. In this way the average transport per second through a cross section is determined. The number of waves in each test varied between 250 waves and 900 waves. Van der Meer and Veldman (1992) A 3D laboratory investigation at two different berm breakwater geometries was carried out at Delft Hydraulics Laboratory by van der Meer and Veldman (1992). LT was measured by the movement of stones from a coloured band for developed profiles. In particular, the measured LT was defined as the number of stones that was displaced per wave. For practical cases, multiplication of this quantity with the storm duration (the number of waves) would lead to a transport rate of total number of stones displaced per storm. Subsequently, the transport rate can be calculated in stones/wave or m 3/s. Eq. (8) valid for the LT estimation at gravel beaches was derived during these experiments. DHI (1995) As part of the EU-funded MAST2 programme on Berm Breakwater Structures, in 1995 3D model tests were carried out at the Danish Hydraulics Institute wave basin (Alikhani et al., 1996; DHI, 1995). The tests were intended to study the behaviour of a reshaped structure under the attack of oblique waves. One of the main objectives of the investigation was to obtain more information on reshaping and longshore transport at berm breakwaters trunks and roundheads. LT rates have been determined by means of visual observations of the longshore transported stones (counting the number of moved coloured stones) from a 1 m wide strip at the trunk and 10° angle sectors at the roundhead; video recording of stone movements during all the tests have been also performed.

33

was equipped with three load cells, which were capable of weighing the amount of trapped sand. The total LT rate was computed by summing the volume of sediment measured by all of the traps. 5. Model verification The full database consisting of a large quantity of observations from field and laboratory experiments has been adopted to verify the ability of the proposed general model in calculating the LT rate for a wide mobility range of the units: from stones to sands. In particular, three types of coastal structures have been considered in the comparisons: reshaping berm breakwaters, gravel and sandy beaches. Fig. 2 shows the relationship between the calculated LT rate in number of units per wave, SN/sin θk,b, versus N⁎⁎ s . Data are presented in a log–log scale. With reference to the range of variation of N⁎⁎ s , three different regions can be observed. The first region, with 2 b N⁎⁎ s b 6, refers to the berm breakwaters; the second refers to the gravel beaches ⁎⁎ (9 b N⁎⁎ s b 23); the third considers sandy beaches (Ns > 23). The region with 6 b N⁎⁎ s b 9 may be considered as a transition region referring to rock slopes and cobbles beaches; for this case not much experience may be found and laboratory and field investigations are needed. Data points in the range of sandy beaches present a higher dispersion in comparison with data in the regions where a lower mobility level is encountered. In Fig. 3 the calculated against observed values of SN/sin θk,b have been plotted showing a satisfactory agreement. In general, the agreement between experimental data and calculated results is good although a certain overestimation is found at low mobility conditions (2 b N⁎⁎ s b 6). In the present study, a quantitative analysis regarding the scatter of the calculated LT rates in terms of volume per second, QLT,c, with respect to the observed values, QLT,o, has been performed. As a measure of the scatter, the root-mean-square error, srms, has been estimated according to Bayram et al. (2007):

srms ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N  2 uX u logQ LT;c − logQ LT;o u t i¼1 N−2

ð23Þ

Large-Scale Sediment Transport Facility (LSTF) (Smith et al., 2003) The 3D experiments were conducted at the LSTF in the Coastal and Hydraulics Laboratory of the U.S. Army Engineer Research and Development Center in Vicksburg, Mississippi (Smith et al., 2003, Wang et al., 2002). Sediment fluxes were obtained from twenty 0.75 wide traps situated at the downdrift end, which covered the entire surf zone. Eighteen traps were installed in the downdrift flow channels and two additional traps were located landward of the first flow channel to measure LT rate across the surf zone including the swash zone. Each trap

where N represents the number of data; a small srms value implies a small scatter. In addition, a mean discrepancy ratio, dr, has been assigned to the LT model given by the percentage of the QLT,c points within an interval of confidence between 0.5 and 2 of QLT,o; this value is substracted from 100% to yield a small number for good agreement. An extended dr within an interval of 0.25 to 4 of the observation points has been also considered in the analysis. Fig. 4

Fig. 2. Calculated SN/sin θk,b according to Eq. (11) against N⁎⁎ s .

Fig. 3. SN/sin θk,b calculated according to Eq. (11) vs SN/sin θk,b observed.

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G.R. Tomasicchio et al. / Coastal Engineering 71 (2013) 28–36 Table 6 Summary of the predicted capability of all the investigated formulae.

Fig. 4. QLT,c according to Eq. (22) vs QLT,o.

shows the comparison between the calculated against observed values of QLT together with the two considered intervals of confidence for dr. Data for low mobility present the larger scatter. The difference is particularly relevant for the five points referring to van der Meer and Veldman (1992) data with 50° obliquity. A preliminary motivation for the discrepancy is related to the observation that, for these high obliquity tests, offshore equivalent conditions do not exist: since the procedure used at that time for the evaluation of breaking conditions referred to equivalent offshore conditions, it was supposed that this could be the reason of the abrupt discrepancy. The proposed new procedure does not assume the existence of offshore conditions and the discrepancy persists. With regard to the scatter from the DHI (1995) data set, a motivation can be the adopted long reshaping phase where displacement and transport are very sensitive to the armouring phenomena which are not accounted for the proposed model. Table 5 summarizes the estimations of srms and dr for the different range of Ns** investigated. typical of reshaping berm breakwaters and In the ranges of N⁎⁎ s sandy beaches a quite similar behaviour is observed: a small value of srms results in both cases and an appreciable percentage of the QLT,c points are within a factor of 0.5 to 2 of the QLT,o. A lower discrepancy is obtained within the extended interval of dr between 0.25 and 4 where a much larger number of QLT,c points are included. In the case of gravel beaches, QLT,c against QLT,o gives the lowest estimations of srms and dr; this behaviour is probably due to a reduced number of data taken into account, equivalent to 4% of the entire database.

N⁎⁎ s

Formula

srms

Data with dr between 0.5 and 2 (%)

Data with dr between 0.25 and 4 (%)

2–6 9–23 >23

van der Meer and Veldman (1992) van der Meer (1990) CERC (USACE, 1984)

2.39 0.39 1.69

61.9 0.0 66.7

45.2 0.0 43.6

Comparison of the results summarized in Tables 5 and 6 reveals that the proposed LT model exhibits the smallest srms and dr compared to the other three formulae followed by, in order, van der Meer (1990) (Eq. (7)), CERC (USACE, 1984) (Eq. (1)) and van der Meer and Veldman (1992) (Eq. (8)). The formula by van der Meer and Veldman (1992) for the case of reshaping berm breakwaters gives the largest srms and dr; the formula by van der Meer (1990) produced somewhat better predictions than the other two considered formulae. Fig. 5 shows the flowchart of the proposed LT model to guide the users in the practical applications. 7. Concluding remarks The present manuscript proposes a general model to determine the total LT at coastal bodies accounting for a large number of mobility conditions of the units composing the mound: from stones to sands. The new model aims to represent an engineering single tool allowing to determine the LT rate at any given coastal mound. It considers an energy flux approach combined with an empirical/statistical relationship between the wave induced forcing and the number of moving units. As in the case of well popular formulae (e.g. CERC formula), the new model does not distinguish between bed load and suspended transport. Calibration and verification of the new LT model have been conducted with nine high quality data sets from field and laboratory observations. The predictive capabilities of the model have also been favourably verified against some of the existing formulae

6. Predictive capability The predictive capability of the new LT model has also been verified against the most popular formulae adopted for the calculation of the LT rate at reshaping berm breakwaters, gravel and sandy beaches. In particular, comparisons have been performed by using the formula of van der Meer and Veldman (1992) with data with N⁎⁎ in the range from 2 to 6, van der Meer (1990) with N⁎⁎ in the s s range from 9 to 23 and the CERC (USACE, 1984) formula with N⁎⁎ s > 23. The estimated srms and dr values for all the investigated formulae are listed in Table 6.

Table 5 Summary of the predicted capability of the new LT model. N⁎⁎ s

srms

Data with dr between 0.5 and 2 (%)

Data with dr between 0.25 and 4 (%)

2–6 9–23 >23

1.52 0.23 1.52

60.0 0.0 65.3

28.3 0.0 40.0

Fig. 5. User flowchart.

G.R. Tomasicchio et al. / Coastal Engineering 71 (2013) 28–36

adopted for the calculation of the LT rate at different coastal mounds. The estimated srms and dr values show that the proposed model gives a better agreement with the observed data with respect to other investigated formulae, for all types of considered coastal bodies: berm or reshaping breakwaters, rock and cobbles beaches, gravel and sandy beaches. The new LT model joined to an accurate description of local wave conditions can represent a robust and easy to use instrument to design a coastal structure. 8. Notations a c cg cg,o cg,b ck,b dr Dn50 D90 E F g h hb H0 Hb Hk Hk,b Hs Hs,0 Hs,b Hs,d H1/50 H2% k k0 K L ld mb Nod Ns N⁎⁎ s p QLT QLT,c QLT,m QLT,o s0 sm,0 sm,k sp srms SN T Tm Tp Ts Tp⁎ V ws

= porosity index [−] = wave celerity [m/s] = wave group celerity [m/s] = offshore wave group celerity [m/s] = wave group celerity at breaking [m/s] = characteristic wave celerity at breaking [m/s] = mean discrepancy ratio [−] = nominal diameter of the unit [m] = 90% representative grain diameter [m] = wave energy [N/m] = wave energy flux [N/s] = gravity acceleration [m/s 2] = water depth [m] = water depth at breaking [m] = off-shore wave height [m] = wave height at breaking [m] = characteristic wave height [m] = characteristic wave height at breaking [m] = significant wave height [m] = off-shore significant wave height [m] = significant wave height at breaking [m] = local significant wave height [m] = mean wave height of the 1/50 fraction of the highest waves [m] = wave height exceeded by 2% of waves [m] = wave number [m −1] = off-shore wave number [m −1] = empirical coefficient [−] = wave length [m] = length of displacement [m] = beach slope at breaking [−] = number of displaced particles at the end of 1000 waves attack [−] = stability number [−] = modified stability number (Lamberti and Tomasicchio, 1997) [−] = proportionality constant [−] = LT rate in volume per unit time [m 3/s] = LT rate in volume per unit time, calculated [m 3/s] = LT rate of the immersed mass per unit time [kg/s] = LT rate in volume per unit time, observed [m 3/s] = off-shore wave steepness = 2πH0/gT 2 [−] 2 = off-shore mean wave steepness = 2πHs,0/gTm [−] 2 = characteristic mean wave steepness = 2πHk/gTm [−] 2 = peak wave steepness = 2πHs/gTp [−] = root-mean-square error [−] = LT measured as number of units per wave [−] = wave period [s] = mean wave period [s] = peak wave period [s] = significant wave period [s] = dimensionless wave period [−] = mean longshore current velocity [m/s] = fall speed [m/s]

γb Δ ε θ θb θd θk,b θ0 ρs ρ ∝

35

= breaker index [−] = relative mass density of the unit = (ρs – ρ)/ρ [−] = transport coefficient [−] = wave angle [°] = wave angle at breaking [°] = displacement angle [°] = characteristic wave angle at breaking [°] = off-shore wave angle [°] = mass density of the unit [kg/m 3] = mass density of water [kg/m 3] = y ∝ x means that y is proportional to x [−]

Subscripts b = at wave breaking point k = characteristic value m = relative to spectrum mean frequency 0 = in offshore conditions p = relative to spectrum peak frequency

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