L. Erikson et al. / Coastal Engineering 52 (2005) 285302
289
mu=yhulsq and mb=yhblsq, and the momentums are
with the mean up-rush or back-wash velocity from
Eq. (2). For swash conditions where Hb0.8, the
Mu=muusu(t) and Mb=mbusb(t), for the up-rush and
bLaw of the WallQ friction factor, Eq. (7), is used and
back-wash, respectively. A new velocity following
collision, ucoll, is determined from the average of the
for conditions where Hz0.8, the sediment-laden
s
friction factor, Eq. (8), is solved iteratively. An
up-rush and back-wash momentums and assuming
iterative approach is necessary due to the shear stress
that momentum is conserved,
term in the denominator which is a function of f as
M coll
ucoll
for M coll N 0
well. The approach is somewhat computationally
x
mb
inefficient in that the swash front velocities are first
calculated using a clear-fluid formulation for the
M coll
friction factor, and if it is found that the value of the
ucoll
for M coll b 0
10
x
mu
Shield's parameter, itself a function of the friction
factor, exceeds 0.8, the swash front velocities and
where
time trajectories are recalculated using the sediment-
Mu Mb
laden friction factor. This approach is used in this
M coll
11
:
2
model to allow for a theoretical estimate of the
friction term, f, as opposed to using an empirical
For the formulation presented here, mb b mu (see
value. The method allows for different up-rush and
Section 5.2) and hence the conditions of negative or
positive values of Mcoll in Eq. (10) were set.
that they are constant during the swash phase in
The new velocity after collision (Eq. (10)) may be
negative and hence the u-subscript is not specified on
question.
ucoll. A negative value would indicate that the up-rush
s
is fully drowned by the returning back-wash.
3.2. Swash interaction
Swash interaction at the SWS is explicitly
The model accounts for two processes in the
accounted for in the model by imposing an exponen-
interaction between successive bores within the swash
tially decreasing velocity of the returning back-wash
zone above the SWS. The first is dcatch-up and
at the SWS. Velocities calculated with Eq. (2) go to
absorptionT where the front of a wave moving
zero after the swash front passes the SWS causing a
discontinuity in the model as the back-wash
landward (up-rush) is passed by a subsequent bore
ddisappearsT at this point. If the velocity is allowed
moving in the same direction (Fig. 1a). The model
to go to zero at the SWS, initial shoreline velocities
simulates the position and velocity of the leading edge
may be overestimated since they are not measured
of the swash (i.e., Eqs. (5) and (2)), and so, in the case
of catch-up and absorption, the model is written to
directly but are calculated with measured wave
follow the faster swash front, effectively drowning the
heights at the SWS (Eq. (3)); any retarding effect
first but slower up-rush. The second process is
that the back-wash may have on the initial up-rush
dcollisionT whereby two separate fronts collide as the
velocity will not be included if the back-wash is
allowed to go to zero at the SWS. The decreasing
back-wash of a preceding swash lens meets the front
velocity at the SWS immediately after the waves pass
of a subsequent swash wave during its up-rush phase
is described by u(t)=uswseat, where t=0 as the swash
as depicted in Fig. 1b. For such a case, a new leading
front passes the SWS, usws is the velocity at the SWS
edge velocity is calculated based on the principals of
at t=0, a=sin(b)/hs and hs is the water depth at the
momentum.
The momentum (mass times velocity) is calculated
boundary of the surf and swash zones (arbitrarily set
for both the up-rush and back-wash at the point (and
at 0.03 m for these simulations).
time) where the fronts meet. To calculate the
3.3. Potential for swash interaction
momentum it is assumed that a fluid element of
length ls at the leading edge of the up-rush collides
The potential for interaction between subsequent
with a fluid element of the same length, at the leading
swash waves on the beach above the SWS may be
edge of the back-wash. The mass of each are
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