DTIC ADA578445: A Unified Sediment Transport Model for Inlet Application

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Journal of Coastal Research SI 59 27-38 West Palm Beach, Florida 2011

A Unified Sediment Transport Model for Inlet Application

Magnus Larson†, Benoît Camenen‡, and Pham Thanh Nam†
†Department of Water Resources Engineering ‡Cemagref
Lund University HHLY, 3 bis quai Chauveau
Box 118 CP 220 F-69336 www.cerf-jcr.org
S-22100 Lund, Sweden Lyon cedex 09, France

ABSTRACT

LARSON, M.; CAMENEN, B., and NAM, P.T., 2011. A Unified Sediment Transport Model for Inlet Applications.
In: Roberts, T.M., Rosati, J.D., and Wang, P. (eds.), Proceedings, Symposium to Honor Dr. Nicholas C. Kraus,
Journal of Coastal Research, Special Issue, No. 59, pp. 27-38. West Palm Beach (Florida), ISSN 0749-0208.

Robust and reliable formulas for predicting bed load and suspended load were developed for application in the
nearshore zone where waves and currents may transport sediment separately or in combination. Also, a routine was
included to determine the sediment transport in the swash zone, both in the longshore and cross-shore directions. An
important objective of the development was to arrive at general sediment transport formulas suitable for a wide range
of hydrodynamic, sedimentologic, and morphologic conditions that prevail around coastal inlets. Thus, the formulas
yield transport rates under waves and currents, including the effects of breaking waves, wave asymmetry, and phase
lag between fluid and sediment velocity for varying bed conditions. Different components of the formulas were
previously validated with a large data set on transport under waves and currents, and in the present paper additional
comparisons are provided for the complete formulas using data on longshore and cross-shore sediment transport from
the laboratory and the field, encompassing the offshore, surf, and swash zones. The predictive capability of the new
formulas is the overall highest among a number of existing formulas that were investigated. The complete set of
formulas presented in the paper is collectively denoted the Lund-CIRP model.

ADDITIONAL INDEX WORDS: Bed load, suspended load, swash zone, waves, current, coastal inlets,
mathematical model, transport formulas.

INTRODUCTION swash motion and transport on the foreshore. The wind and tide
generate mean circulation patterns that move sediment,
Many sediment transport formulas have been developed especially in combination with waves. Also, on the bay side and
through the years for application in the coastal areas (Bayram et in the vicinity of the inlet throat, river discharge to the bay might
al., 2001; Camenen and Larroude, 2003). However, these generate currents that significantly contribute to the net
formulas have typically focused on describing a limited set of transport. Figure 1 illustrates some of the hydrodynamic forcing
physical processes, which restrict their applicability in a around an inlet that is important for mobilizing and transporting
situation where many processes act simultaneously to transport sediment.
the sediment, for example, around a coastal inlet. Also, many of Predicting sediment transport and morphological evolution
the formulas have not been sufficiently validated towards data, around an inlet is crucial for the analysis and design of different
but they have typically been calibrated and validated against engineering activities that ensure proper functioning of the inlet
limited data sets. Thus, there is a lack of general sediment for navigation (see Figure 2). Optimizing dredging operations
transport formulas valid under a wide range of hydrodynamic, due to channel infilling or minimizing local scour, which may
sedimentologic, and morphologic conditions that yield reliable threaten structural integrity, are examples of such activities.
and robust predictions. In this paper such formulas are presented Furthermore, bypassing of sediment through the inlet shoals and
and validated against high-quality laboratory and field data on bars are vital for the supply of material to downdrift beaches and
longshore and cross-shore sediment transport. any reduction in this transport may cause severe erosion and
The coastal environment around an inlet encompasses shoreline retreat. After an inlet opening, as the shoals and bars
hydrodynamic forcing of many different types, where waves, grow with little bypassing transport, downdrift erosion is
tides, wind, and river runoff are the most important agents for common and varying engineering measures such as beach
initiating water flows and associated sediment transport. Besides nourishment and structures might be needed. On the updrift side
the oscillatory motion, waves induce mean currents in the surf accumulation normally occurs, especially if the inlet has been
zone (longshore currents, rip currents, etc.), stir up and maintain stabilized with jetties, with shoreline advance and increased
sediment in suspension through the breaking process, and cause infilling in the channel.
Considering the inlet environment, a general sediment
____________________
DOI: 10.2112/SI59-004.1 received 8 October 2010; accepted 3 May transport model should yield predictions of the transport rate
2010. taking into account the following mechanisms:
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14. ABSTRACT
Robust and reliable formulas for predicting bed load and suspended load were developed for application in
the nearshore zone where waves and currents may transport sediment separately or in combination. Also,
a routine was included to determine the sediment transport in the swash zone, both in the longshore and
cross-shore directions. An important objective of the development was to arrive at general sediment
transport formulas suitable for a wide range of hydrodynamic, sedimentologic, and morphologic conditions
that prevail around coastal inlets. Thus, the formulas yield transport rates under waves and currents,
including the effects of breaking waves, wave asymmetry, and phase lag between fluid and sediment
velocity for varying bed conditions. Different components of the formulas were previously validated with a
large data set on transport under waves and currents, and in the present paper additional comparisons are
provided for the complete formulas using data on longshore and cross-shore sediment transport from the
laboratory and the field, encompassing the offshore, surf, and swash zones. The predictive capability of the
new formulas is the overall highest among a number of existing formulas that were investigated. The
complete set of formulas presented in the paper is collectively denoted the Lund-CIRP model.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 12
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Prescribed by ANSI Std Z39-18
28 Larson, Camenen, and Nam
_________________________________________________________________________________________________

In this paper, general formulas are presented to compute the
sediment transport rate in an inlet environment that includes all
of the above mechanisms. These formulas are mainly based on
previous work by the authors (see Camenen and Larson, 2005;
2006; 2008), where different components of the formulas were
developed, calibrated, and validated against extensive data sets.
In the following the complete set of formulas is collectively
denoted as the Lund-CIRP model. The primary objective of this
study was to develop robust and reliable sediment transport
formulas applicable under a wide range of conditions
encountered at a coastal inlet, and to validate these formulas
towards high-quality sediment transport rate measurements
obtained in the laboratory or in the field.
This paper is organized as follows. The formulas developed to
compute sediment transport are first described. Validation of the
bed load and suspended load formulas, as well as five other

Figure 1. Hydrodynamic processes controlling the sediment transport in existing formulas, was carried out based on laboratory and field
an inlet environment (from Camenen and Larson, 2007). measurements of the longshore and cross-shore sediment
transport in the surf and offshore zone. Then, the transport in the
swash zone was validated using laboratory data on the longshore
transport rate. Finally, the conclusions are presented.

SEDIMENT TRANSPORT MODEL

Bed Load Transport

Camenen and Larson (2005) developed a formula for bed load
transport based on the Meyer-Peter and Müller (1948) formula.
The bed load transport (q ) may be expressed as follows,
sb

q   
sbw a   expb cr 
(s1)gd530 w net cw,m  cw (1)
q   
sbn a   expb cr 
(s1)gd530 n cn cw,m  cw

where the subscripts w and n correspond, respectively, to the
wave direction and the direction normal to the waves, s ( /)
s
is the specific gravity of the sediment, in which  is the density
s

of the sediment and ρ of the water, g the acceleration due to
Figure 2. Engineering activities around an inlet for which predictions of
sediment transport and morphological evolution are of importance (from gravity, d50 the median grain size, aw, an, and b are empirical
Camenen and Larson, 2007). coefficients (to be discussed later),  the critical Shields
cr
number for initiation of motion (obtained from Soulsby, 1997),

 the mean Shields number and  the maximum Shields
cw,m cw
Bed load and suspended load number due to wave-current interaction, and
Waves and currents   f U sin2/(s1)gd /2 (where fc is the current-
cn c c 50
Breaking and non-breaking waves
related friction factor, U the steady current velocity, and  the
c
Slope effects
angle between the wave and the current direction). In order to
Initiation of motion
simplify the calculations, the mean and maximum Shields
Asymmetric wave velocity
numbers due to wave-current interaction are obtained by vector
Arbitrary angle between waves and current
addition:  22 2  cos1/2 and
Phase-lag effects between water and sediment motion cw,m c w,m w,m c
Realistic bed roughness estimates (e.g., bed features)  22 2  cos1/2, where  , , and  are the
Swash-zone sediment transport cw c w w c c wm w
current, mean wave, and maximum wave Shields number,
respectively and   /2 for a sinusoidal wave profile.
w,m w
Journal of Coastal Research, Special Issue No. 59, 2011
A Unified Sediment Transport Model for Inlet Applications  29
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The net sediment transporting Shields number  in Eq. (1)
net
(a) (b)
is given by, u(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)
  Uw,max (cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)
net (1pl,b)cw,on(1pl,b)cw,off   (2) Uccosϕ (cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)Uw,max
(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2) 2Uw
  ϕ (cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)
where cw,on and cw,off are the mean values of the instantaneous Uc (cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)U(cid:0)(cid:0)(cid:2)(cid:2)cc(cid:0)(cid:0)(cid:2)(cid:2)os(cid:0)(cid:0)(cid:2)(cid:2)ϕ(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:0)(cid:2)(cid:2) t
shear stress over the two half periods Twc and Twt (Tw=Twc-Twt, in Uw,min Ucsinϕ Twc (cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)T(cid:0)(cid:2)w(cid:0)(cid:2)t (cid:0)(cid:2)(cid:0)(cid:2)(cid:0)(cid:2)
which Tw is the wave period), and pl,b a coefficient for the Tw
Figure 3. Definition sketch for current and wave direction and for
phase-lag effects (Camenen and Larson, 2006). In the same way
bottom velocity profile in the direction of wave propagation (from
as for the Dibajnia and Watanabe (1992) formula, the mean
Camenen and Larson, 2007).
values of the instantaneous shear stress over a half period are

defined as follows (see Figure 3),

  1 Twc fcw(Uccosuw(t))2dt    Wh
cw,on Twc 0 2(s1)gd50 (3) qsswUc,netcRW 1exp s  (6)
s
  1 Tw fcw(Uccosuw(t))2dt    Wh
cw,off Twt Twc 2(s1)gd50 qssnUcsincRW 1exp s 
s

where u (t) is the instantaneous wave orbital velocity, t time,
w where h is the water depth, U the net mean current after a
and f the friction coefficient due to wave-current interaction c,net
cw wave period, c the reference concentration at the bottom, W the
introduced by Madsen and Grant (1976), R s
sediment fall speed, and  the sediment diffusivity. The ratio

Wh/ may often be assumed large, implying that the
f  X f (1X )f (4) s
cw v c v w
exponential term is close to zero. However, such an assumption

may not be valid when strong mixing due to wave breaking is
with Xv Uc/(UcUw), where fw is the wave-related friction present. The bed reference concentration is obtained from,
factor, U the mean current velocity, and U the average of the
c w
peak velocities during the wave cycle (the root-mean-square
  
(rms) value is used for random waves). c A  exp4.5 cr  (7)
Based on comparison with an extensive data set (Camenen R cR cw,m  cw
and Larson, 2005), the following relationship is proposed for the
transport coefficient aw, in which the coefficient AcR is given by,

a 66X (5) A 3.5103exp(0.3d ) (8)
w t cR *

in which Xt c/(cw) . The coefficient for transport where d 3(s1)g/2d is the dimensionless grain size and 
* 50
perpendicular to the waves, where only the current moves
the kinematic viscosity.
sediment, is set to a=12, and the coefficient in the term
n The sediment diffusivity is related to the energy dissipation
describing initiation of motion is b=4.5. The phase-lag effects
(Battjes and Janssen, 1978),
are introduced through the coefficient   
pl,b on off
following Camenen and Larson (2006; detailed coefficient 1/3
D
expressions not given here).   h (9)

Suspended Load Transport
in which D is the total effective dissipation expressed as,
In determining the suspended load q , following the simplified
ss
approach by Madsen (1993) and Madsen et al. (2003), the Dk3D k3D k3D (10)
b b c c w w
vertical variation in the horizontal velocity was neglected and an

exponential-law profile assumed for the sediment concentration .
where the energy dissipation from wave breaking (D) and from
b
Camenen and Larson (2007) made a comparison between
bottom friction due to current (D) and waves (D ) were simply
c w
representing the velocity variation over the vertical in the
added, and k , k, and k are coefficients. The coefficient k
b c w b
sediment transport calculations and using the average velocity,
corresponds to an efficiency coefficient related to wave
finding a small difference in the obtained total suspended load.
breaking, whereas k and k are related to the Schmidt number
c w
Thus, the suspended sediment load may be obtained from
(Camenen and Larson, 2007). The mean value for the vertical
(Camenen and Larson, 2008),
Journal of Coastal Research, Special Issue No. 59, 2011
30 Larson, Camenen, and Nam
_________________________________________________________________________________________________

sediment diffusivity employed in the exponential concentration The bottom slope may influence the sediment transport,
profile may be determined by integrating the vertical variation in especially if it is close to the critical value given by the wet
the diffusivity (Camenen and Larson, 2007). Assuming a internal friction angle of the sediment. In order to take into
parabolic profile for this variation (Dally and Dean, 1984), the account the local slope, the transport rate may be multiplied with
mean value over the depth (for a steady current or waves, a function containing the local slope and a coefficient,
respectively) may be written as follows,
 z 
 Dj 1/3hk u h (11) qs* qs1 sb  (16)
j    j *j
where β is a coefficient for the slope effects (0.5<β<2) and

∂z/∂s is the local slope. Bailard (1981) presented values on the
where k is a function of a non-dimensional number σ b
j j coefficient β that depends on the sediment transport regime.
expressing the ratio between the vertical eddy diffusivity of

particles and the vertical eddy viscosity of water (inverse of the
Swash Zone Transport
common definition of the Schmidt number), and u is the shear
∗,j
velocity due to current or waves only with subscript j taking on
Larson et al. (2004) and Larson and Wamsley (2007) developed
the values c (current) or w (waves), respectively. In case of a
formulas for the net transport rate in the swash zone,
steady current, k =σ/6 ( = 0.41 is von Karman’s constant),
c c
whereas for waves k =σ /3. The following expression was
w w tan u3dh t
d eveloped, qbc,net Kc tan2 dmh/dx2 g0dxtaneT0 (17)
m
W  W tan u2v t
  A A sin2 s  s 1 q K m 0 0 0
j j1 j2 2u  u bl,net l tan2 dh/dx2 g T
 *j  *j m

W  W
 1A A 1sin2 s  s 1 where qbc,net and qbl,net are the net transport rates over a swash
j j1 j2 2u  u cycle in the cross-shore and longshore directions, respectively,
 *j  *j
K and K are empirical coefficients, ϕ the friction angle for a
(12) c l m
moving grain, β the foreshore equilibrium slope, u, v the
e 0 0
scaling velocities (cross-shore and longshore directions,
where j is a subscript equal to c or w, and A =0.4 and A =3.5 or
c1 c2 respectively) and t the scaling time, and T the swash duration
A =0.15 and A =1.5. For wave-current interaction, a weighted 0
w1 w2 (taken to be similar to the incident wave period). Ballistics
value is employed:
theory, neglecting friction, may be employed to compute swash

zone hydrodynamics including the scaling parameters and their
cwXtc1Xtw (13) variation across the foreshore (see Larson and Wamsley, 2007).
The interaction between uprush and backwash processes in
The net mean current is defined in a similar way to the net the vicinity of the shoreline induces considerable sediment
Shields number for the bed load in order to take into account a stirring and movement, which is of importance to describe when
possible sediment transport due to wave asymmetry, as well as the sediment exchange between the swash zone and the inner
possible phase-lag effects on the suspended concentration, surf zone is modeled. Since horizontal advection and diffusion
of sediment is particularly significant in this region, it may be
U 1 U 1 U (14) necessary to include those processes in a model instead of
c,net pl,s cw,on pl,s cw,off calculating local transport rates under the assumption that
horizontal sediment exchange is negligible. By solving the
where αpl,s is the coefficient describing phase-lag effects on the advection-diffusion (AD) equation for the sediment
suspended load (Camenen and Larson, 2006; detailed coefficient concentration (suspended load), a more realistic description of
expressions not given here), and Ucw,j is the rms value of the the horizontal mixing is obtained in the inner surf zone.
velocity (wave + current) over the half period Twj, where the The two-dimensional time- and depth-averaged AD equation
subscript j should be replaced either by on (onshore) or off is expressed for steady-state conditions as,
(offshore) (see also Figure 3) according to:
   
Ucw,on T1 0Twc(Uccosuw(t))2dt (15)  Cxqx  Cyqy xKxhCxxKyhCyPS (18)
wc
U  1 Tw(U cosu (t))2dt where Cis the depth-averaged sediment concentration, qx and qy
cw,off Twt Twc c w are the flow per unit width parallel to the x and y axes,
respectively, Kx and Ky are the sediment diffusion coefficients in
In case of a steady current U = U. x and y direction, respectively, P is the sediment pick-up rate,
c,net c
Journal of Coastal Research, Special Issue No. 59, 2011
A Unified Sediment Transport Model for Inlet Applications  31
_________________________________________________________________________________________________
and S is the sediment deposition rate. From Elder (1959), the APPLICATION OF SEDIMENT TRANSPORT
sediment diffusion coefficient is estimated as, FORMULAS

K K 5.93u h (19) Background and Data Employed
x y *c

Various components of the Lund-CIRP model presented in
where u is the shear velocity from the current only. This the previous section were validated in earlier studies against
*c
equation was used by Buttolph et al. (2006) to describe sediment extensive data sets on sediment transport, and empirical
diffusion in Eq. 18, although Elder (1959) derived his expressions were derived for the most important coefficient
expression based on studies on longitudinal dispersion in a values. These data sets consisted mainly of laboratory
channel. However, qualitatively good results have been achieved experiments, although field data were also included to some
with this formulation for a number of test cases. Buttolph et al. extent. Camenen and Larson (2005, 2006, 2008) presented the
(2006) also included additional mixing due to waves in the results of the comparison for current, waves, and wave-current
sediment diffusion coefficient, but this was not done here. interaction. Both sinusoidal and asymmetric waves were
The sediment pick-up and deposition rates, respectively, are included, as well as non-breaking and breaking waves. Selected
given by, existing formulas, commonly used in engineering studies, were
also employed to compute the transport rates and comparison
Pc W (20) with the Lund-CIRP model showed that overall this model
R s displayed the best agreement with data. The swash zone

transport formula was validated by Larson et al. (2004) and
C Larson and Wamsley (2006) with regard to the cross-shore and
S  W (21)
 s longshore components, respectively.
d In the following, the complete formulas are compared to data

from the laboratory and the field where measurements were
where  is a coefficient calculated based on Camenen and simultaneously collected at several locations along a profile
d
Larson (2008; see also Buttolph et al., 2006): under realistic wave and current conditions. Data on the
longshore sediment transport rate from the Large-scale Sediment
Transport Facility (LSTF) at the Coastal and Hydraulics
   Wh
d  wh1exp s  (22) Lfieabldo reaxtpoeryri m(CenHtLs )a ti nth eV Ficikelsdb uRregs, eMarcishs iFsasicpilpiit,y a(nFdR Ffr)o omf CseHvLer ianl
s
Duck, North Carolina, were employed. Also, data on the cross-
shore sediment transport rate were used from the Duck field
Simplified Transport Formulas
experiments. Previously the various components of the Lund-
The Lund-CIRP model was developed to describe a wide CIRP model were validated mainly with point values, often
range of different processes occurring around a coastal inlet. under a limited set of hydrodynamic forcing and under
Thus, to apply the complete formulas might be time-consuming conditions that are not completely analogous to a natural beach
or require extensive background data not always available. In (e.g., oscillatory water tunnels). Thus, the present testing
many cases satisfactory results can be achieved with simplified constitutes a more general validation of the complete formulas.
versions of the formulas where certain processes or phenomena A number of commonly utilized transport formulas were also
are not included. For example, wave asymmetry is often employed to compute the transport rate, including the formulas
important immediately seaward of the surf zone, but in a of Bijker (1968), Bailard (1981), van Rijn (1989), Watanabe
regional perspective the asymmetry may be neglected with little (1992), and Dibajnia and Watanabe (1992). These formulas
loss in simulation results. By not including wave asymmetry were selected because they are often used in numerical models
gains are made in calculation speed, since the integration over of the morphological evolution. It should be noted that similarly
the wave cycle may be greatly simplified. Also, the difficulties to the present formula the Bijker, Bailard, and van Rijn formulas
in estimating the asymmetry over large spatial areas are avoided. estimate the bed load and suspended load separately, whereas
Most regional wave models used in simulating the the Watanabe and Dibajnia/Watanabe formulas directly estimate
morphological evolution rely on linear wave theory and do not the total load. The Bailard, Dibajnia/Watanabe, and Lund-CIRP
yield any information on the wave asymmetry. Another model take into account the effects of wave asymmetry on the
phenomenon that is often neglected is the phase lag between total sediment transport (a large portion of the transport rates
water flow and sediment movement, which may have effects on derived from the measurements do not include this effect).
the net sediment transport rate over a wave cycle. The Furthermore, in the Dibajnia/Watanabe and the Lund-CIRP
importance of the phase lag depends on the hydrodynamic and model the phase-lag effects are taken into account (again, this is
sediment characteristics. Thus, a simplified version of the not done in the transport rates derived from the measurements).
transport formulas would employ Eqs. 1 and 6 with only the An important quantity for many of these formulas is the
current contributing to the net sediment transporting velocity. Shields parameter. The roughness height was estimated using
the Soulsby (1997) method, which includes roughness
contributions from sediment grains (i.e., skin friction), bed
forms, and sediment transport. The ripple characteristics were
Journal of Coastal Research, Special Issue No. 59, 2011
32 Larson, Camenen, and Nam
_________________________________________________________________________________________________

estimated using the Grasmeijer and Kleinhans (2004) equations,
and roughness due to sediment transport was obtained from the
formula by Wilson (1989). Uncertainties in the final results are m) 0.4 Hrw
to a large degree related to the calculation of the bottom shear (w0.2 w
stress, which in turn depends on the bed roughness. The H
roughness in the presence of ripples is especially difficult to 0
0 5 10 15 20
estimate and there are several formulas available to do this
0.2
(Camenen, 2009). Often a particular sediment transport formula U
has been developed using a specific method to calculate the m/s) 0.1 Ulr
roughness and shear stress. However, in this study the same U ( 0
method was used for all formulas, which may have produced
−0.1
less good agreement with the data for some of the formulas. 0 5 10 15 20
0.5
t
Validation of Longshore Sediment Transport m) 0 t0
(b−0.5 eq
The Lund-CIRP model was validated with measured z −1
longshore sediment transport rates from two data sets, namely a −1.5
0 5 10 15 20
laboratory experiment carried out in the LSTF (Wang et al., a cross−shore distance (m)
2002) and field experiments (SandyDuck) performed at the FRF
(for a summary of the field experiments, see Miller, 1998; x 10−5
1999). For these experiments, the sediment transport rate was ) 5
s measured (traps)
estimated based on the measured time-averaged sediment m/ measured (conc.)
concentration and velocities. Thus, the transport rates mainly 3m/ 4 BBiajkilearrd
rloefalde ctot gtheeth ecru rwreitnht -trheela eteffde cstuss fpreonmd etdh el owaadv, easn odn mthoes tt roanf stphoer tbinegd ort ( DWibaatajnniaab &e Watanabe
velocity are not included (in the concentration measurements sp 3 Van Rijn
n Lund−CIRP
close to the bottom, some portion of the bed load might have a
r
been captured since the gages were close to the bed). Because T
the wave asymmetry, defined as r U /U 1(for notation, ent 2
w w,max w m
see Figure 3), was not recorded, rw was estimated using the di
method proposed by Dibajnia et al. (2001) (examples of e 1
S
calculated results are shown in Figures 4a, 5a, and 7a). For the e
LSTF data set, the cross-shore current (undertow) was not or
h 0
measured, and in order to compute the total shear stress the s
g
undertow was estimated using the model developed by Svendsen n
o
(1984) (Figure 4a). These calculations are a source of error in L−1
0 5 10 15 20
the computation of the suspended load. For the LSTF b Cross−Shore Distance (m)
experiments, measurements of the total load were also carried

out using a trap system at the downdrift end of the basin. A
Figure 4. (a) Cross-shore distribution of hydrodynamic parameters
comparison between the two measurement methods (Figure 4b) together with beach profile shape, and (b) cross-shore distribution of the
indicates the order of magnitude of the uncertainties in the longshore sediment transport rate together with predictions from six
experimental results. It also shows that suspended load is studied formulas for an LSTF case (Test 3).
prevailing.
In Figure 4 are typical results presented for the LSTF
experiment (case with spilling breaking waves). The Watanabe measurements in this region. Thus, a special sediment transport
formula tends to overestimate the transport rates in the surf formula for the swash zone should be included as will be
zone, whereas the van Rijn and Dibajnia/Watanabe formulas discussed later in the paper. Also, the formulas yielded a small
markedly underestimate the rates. The Lund-CIRP model yields negative transport along a limited portion of the profile outside
results of the correct magnitude, even if the peak in the transport the surf zone, since the measured current was negative in this
rate in the outer surf zone is broader and less pronounced region (at one measurement point). The observed sediment
compared to the measured peak. This partly results from the transport did not display any negative values, implying that
computation of the ripple characteristics which are found to be wave-induced sediment transport may prevail over the current-
smaller on the bar and thus induce a smaller bottom shear stress. related transport in this area. Only the Bailard and
Close to the swash zone (still-water shoreline was located at Dibajnia/Watanabe formulas and the Lund-CIRP model could
x=0), all formulas largely underestimate the longshore sediment potentially describe this situation.
transport rate. The influence of the swash zone is significant Figure 5 illustrates typical results obtained for the SandyDuck
near the shoreline, and since the formulas do not include experiments with regard to the longshore sediment transport rate
longshore transport in the swash they fail to reproduce the (case from February 4th 1998). The variation in the data and the
Journal of Coastal Research, Special Issue No. 59, 2011
A Unified Sediment Transport Model for Inlet Applications  33
_________________________________________________________________________________________________
scatter in the predictions are larger than for the LSTF data,
partly because the measurements took place under less 4
H
controlled conditions. In some cases wind was a significant m) 5 wr
factor in generating the current. Most of the formulas predict (w2 w
similar cross-shore variation, but the magnitude differs greatly. H
The Lund-CIRP model, as well as the Bijker formula, tends to 0
0 100 200 300 400 500 600
overestimate the transport rate, whereas the Dibajnia/Watanabe
aFnigdu rBea i5labr)d. fTohrme uWla autanndaebree staimnda teD itbhaej nriaat/eWs a(staenea Tbea bfleo r1m ualnads m/s) 1 U−Ulgcs
yield the total load, so overestimation is expected. Thus, the U (0.5
underestimation by the Dibajnia/Watanabe formula is somewhat 0
0 100 200 300 400 500 600
surprising.
0
Table 1 presents statistical results for the comparison between z
all studied formulas and the experimental data from LSTF (4 m) b
experimental cases encompassing in total 92 data points) and z (b −5
SandyDuck (6 experimental cases encompassing in total 66 data −10
0 100 200 300 400 500 600
points). The table presents the percentage of data correctly a cross−shore distance (m)
predicted within a factor 2 or 5, and the mean value and standard
deviation of the function f(q )logq /q , where qss,pred x 10−3
and qss,meas are the predicted assnd measssu,prreedd vsas,mlueaess, respectively. m/s)10 eBxijpkeerriments
TSSiahmned ilyaWDrlyua,tc aktnh adeb aBeta a ibfloaurrtmd y uaielnaldd sDp priebosaoejrnn itaasg /Wretheameta ennbateb sefto fro rrtehmseuu lLltasS s TsfFoeer dmat thtaoe. 3ort (m/ 8 BDWVaaibainlataa jRrnndiiajanb &e Watanabe
be sensitive to the scale of the experiments. These two formulas sp 6 Lund−CIRP
n
yield overestimation for the laboratory experiment, whereas they a
Tr
underestimate the results for the field experiment (the Watanabe nt 4
formula displays the opposite behavior). This behavior may be e
m
due to the formulas not being a function of the total shear stress di
(which varies with the scale of the experiment), but only of the e 2
S
velocity profile (Bailard and Dibajnia/Watanabe formula) or that e
they are too simple to include all the parameters for the bed load hor 0
and suspended load (Watanabe formula). For the experimental gs
n
cases studied on the longshore sediment transport rate the o
L−2
Bailard formula and the Lund-CIRP model yield the overall best 0 100 200 300 400 500 600
results. b Cross−Shore Distance (m)

As a summary, Figure 6 shows the prediction of the longshore
Figure 5. (a) Cross-shore distribution of hydrodynamic parameters
transport rate (i.e., mainly suspended load) across the beach
together with beach profile shape, and (b) cross-shore distribution of the
profile for all data points from the LSTF and SandyDuck
longshore sediment transport rate together with predictions from six
experiments using the Lund-CIRP model, where the predicted studied formulas for a SandyDuck case (Feb. 4th 1998).
rates were normalized with the measured rates. The plot in

Figure 6a shows the underestimation near the swash zone,
whereas in the zone of incipient breaking transport rates may be Bailard, Dibajnia/Watanabe formula (and the Lund-CIRP model
over- or underestimated. Larger scatter is obtained in the if U is used) allow for a sediment transport in the opposite
c,net
comparison for the SandyDuck data (Figure 6b), together with direction to the mean current, if asymmetric waves are
the overestimation of the transport rates pointed out earlier. prevailing. Onshore sediment transport due to wave asymmetry
often occurs seaward of the point of incipient breaking.
Validation of Cross-shore Sediment Transport Computations with these two latter formulas were performed
with the asymmetry taken into account, although the transport
The cross-shore transport (mainly suspended load) was also rate obtained from the measurements would only include the
available from the SandyDuck experiments. Compared to the transport associated with the mean current (i.e., undertow). If
longshore transport, the predictions of the cross-shore transport U is used, the Lund-CIRP appears to be quite sensitive to the
c,net
with the formulas are often more uncertain because the input balance between undertow and wave asymmetry, indicated by
conditions are less well known due to the complex flow the rapid change in transport direction around the point of
situation. Figure 7 presents some typical results obtained for the incipient breaking. Since the experimental data do not include
experimental case from March 12th 1996. The Bijker, van Rijn, the wave-induced sediment transport, the results differ quite a
and Watanabe formulas and the Lund-CIRP model induce a lot between the formulas outside the surf zone, where the wave
sediment transport which is in the same direction as the asymmetry is strong.
undertow, that is, in the offshore direction. On the contrary, the
Journal of Coastal Research, Special Issue No. 59, 2011
34 Larson, Camenen, and Nam
_________________________________________________________________________________________________

Table 1. Prediction of longshore transport rate for the LSTF and SandyDuck experiments.

Author(s) Pred. x2 Pred. x5 mean(f (qss)) std(f (qss))
LSTF data (4 cross-shore profiles, 92 experiments)
Bijker (1968) 18% 71% 1.4 1.5
Bailard (1981) 20% 75% 1.1 1.4
Van Rijn (1989) 27% 59% −0.8 1.9
Watanabe (1992) 11% 60% 1.4 1.3
Dibajnia & Watanabe (1992) 35% 75% −0.4 1.5
Lund-CIRP 33% 79% 0.8 1.5

SandyDuck data (6 cross-shore profiles, 66 experiments)
Bijker (1968) 35% 85% 0.8 0.5
Bailard (1981) 30% 68% −1.3 0.8
Van Rijn (1989) 20% 48% −1.5 1.2
Watanabe (1992) 61% 91% 0.1 0.9
Dibajnia & Watanabe (1992) 17% 63% −1.4 0.6
Lund-CIRP 39% 85% 0.4 1.0

4
H
m) 5 wr
(w2 w
101 H
0
0 100 200 300 400 500 600
s
a
/ qds,me100 U (m/s) 0.05 U−Ulgcs
e
s,pr exp.1 −0.50 100 200 300 400 500 600
q exp.1 <0 0
exp.3 z
10−1 eexxpp..35 <0 (m)b −5 b
exp.5 <0 z
a exp.6 −10
exp.6 <0 a 0 100 200 300 400 500 600
cross−shore distance (m)
0 5 10 15 20
Cross−Shore Distance (m)
s) 4x 10−3
m/ experiments
9966//0033//1133 <0 3m/ 3.5 BBiajkilearrd
101 999777///000334///330111 <0 port ( 3 DWVaibanata jRnniiajanb &e Watanabe
eas 999777///011400///022100 <<00 Trans 2.25 LLuunndd−−CCIIRRPP with Ucw,net
/ qpreds,m100 99998888////00002222////00004455 <<00 Sediment 1.15
qs, e 0.5
or
h 0
10−1 S
−
s−0.5
s
o
Cr −10 100 200 300 400 500 600
b −100 0 100 200 300 400 500 600 b Cross−Shore Distance (m)
Cross−Shore Distance (m)
Figure 7. (a) Cross-shore distribution of hydrodynamic parameters
Figure 6. Prediction of the longshore transport rate across the profile together with beach profile shape, and (b) cross-shore distribution of the
using the Lund-CIRP model for (a) the LSTF data, and (b) the cross-shore sediment transport rate together with predictions from six
SandyDuck data. studied formulas for a SandyDuck case. (March 12th 1996).

Journal of Coastal Research, Special Issue No. 59, 2011
A Unified Sediment Transport Model for Inlet Applications  35
_________________________________________________________________________________________________
Table 2. Prediction of cross-shore transport rate for the SandyDuck experiments.

Author(s) Pred. x2 Pred. x5 mean(f (qss)) std(f (qss))

SandyDuck data (6 cross-shore profiles, 66 experiments)
Bijker (1968) 14% 41% 2.1 1.2
Bailard (1981) 24% 52% 0.1 1.5
Van Rijn (1989) 26% 55% 0.2 1.6
Watanabe (1992) 47% 68% 0.8 1.0
Dibajnia & Watanabe (1992) 35% 61% 0.0 1.2
Lund-CIRP 24% 67% 1.3 1.0

Table 3. Prediction of the cross-shore transport rate for the sheet-flow experiments by Dohmen-Janssen and Hanes (2002) (∗ denotes
that transport in the opposite direction to the measurement is predicted).

Author(s) Pred. x2 Pred. x5 mean(f (qss)) std(f (qss)) qsb/qss
Bijker (1968) 0%* 0%* -1.1 0.4 0.02
Bailard (1981) 75% 100% -0.4 0.7 0.20
Van Rijn (1989) 0%* 0%* -0.3 0.2 1.4
Watanabe (1992) 0%* 0%* -0.4 0.4 -
Dibajnia & Watanabe (1992) 100% 100% -0.1 0.4 -
Lund-CIRP 100% 100% 0.3 0.6 0.16

Table 2 presents the statistical results for all the formulas
concerning the SandyDuck experiments. The agreement
96/03/13
achieved for the cross-shore transport rate, as indicated by the 96/03/13 <0
table, seems to be poorer than for the longshore transport rate 97/03/31
(Table 1). The Watanabe formula presents the best results, 101 9977//0034//3011 <0
which is surprising since it was calibrated for longshore 97/04/01 <0
transport. However, as discussed previously, the measured s 97/10/20
transport only includes the current-related transport, making the mea 9978//1002//2004 <0
comparisons somewhat biased. s, 98/02/04 <0
In Figure 8 are the predictions of the cross-shore transport / qd100 9988//0022//0055 <0
with the Lund-CIRP model across the profile for all e
pr
experimental cases from SandyDuck shown, where the predicted s,
q
values were normalized with the measurements. The formula
yields an overestimation of the transport in general and a
10−1
significant scatter in the predicted values. This scatter is
however the smallest among the studied formulas.
An interesting data set was provided by Dohmen-Janssen and
Hanes (2002). They measured both bed load and suspended load −100 0 100 200 300 400 500 600
transport in a large wave flume under sheet-flow conditions Cross−Shore Distance (m)

(non-breaking waves). Results were obtained for four
experimental cases, and the results of the comparisons with the Figure 8. Predictions of the cross-shore transport rate across the beach
formulas are presented in Table 3. Although a small undertow profile for the SandyDuck data using the present formula, where the
occurred (opposite to the wave direction), the net sediment predicted values were normalized with the measured values.
transport was directed onshore because of the prevailing

asymmetric waves. The three formulas which assume that the
hydrodynamic and sediment conditions studied. The Bailard
direction of the current determines the direction of the sediment
formula (as well as the Bijker formula) predicts that suspended
transport (Bijker, van Rijn, and Watanabe formulas) predict the
load dominated (q /q =0.02). The Dibajnia and Watanabe
wrong direction for the net total load. The Bailard, sb ss
formula, since it was calibrated for sheet-flow conditions, yields
Dibajnia/Watanabe and the Lund-CIRP model yield the correct
results in good agreement with the observations. Finally, the
direction for the sediment transport, as well as good quantitative
Lund-CIRP model (as well as the Bailard formula) also yields
predictions. Dohmen-Janssen and Hanes (2002) observed that,
good results, but it tends to overestimate the suspended load
in case of sheet flow, bed load was always prevailing and only
(q /q =0.2).
10% of the total load constituted suspended load for the sb ss
Journal of Coastal Research, Special Issue No. 59, 2011