A maior rede de estudos do Brasil

Grátis
45 pág.

## Pré-visualização | Página 10 de 14

```difference could
be used to estimate γL .
Figure 5.14 presents the relationship between initial (θ0) and ﬁnal (θF) swash
angles for varying values of non-dimensional linear friction.
Figure 5.13 Swash trajectories, effect of linear fric-
tion.
P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH
CB373-05 CB373 June 27, 2001 11:28 Char Count= 0
5.6 SWASH ZONE DYNAMICS 119
Figure 5.14 Relationship between initial and ﬁnal swash angles as a function of β (for linear
friction).
5.6.4 NORMALLY INCIDENT WAVES WITH NONLINEAR FRICTION
We commence with Eqs. (5.65) and note for normal incident waves, V ′y = 0. Because
of the absolute value sign in the friction term, it is necessary to consider uprush and
backrush separately.
Uprush – During uprush, V ′ > 0, and the ﬁrst of Eqs. (5.65) becomes
dV ′x
dt ′
= −2− γ V ′2, (5.77)
a solution of which is
V ′(t ′) =
√
2
γ
tan
[√
2γ (t ′max − t ′)
]
(5.78)
in which t ′max is the time at which the maximum particle excursion occurs and is
t ′max =
1√
2γ
tan−1
(√
γ
2
)
(5.79)
The associated uprush displacement is determined by integrating Eq. (5.78) to
yield
x ′(t ′) = 1
γ
�n
{
cos(
√
2γ t ′max)
cos[
√
2γ (t ′max − t ′)]
}
, (5.80)
and the maximum displacement occurs at t ′ = t ′max and is
x ′(t ′max) = x ′max =
1
γ
�n{cos
√
2γ t ′max} (5.81)
P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH
CB373-05 CB373 June 27, 2001 11:28 Char Count= 0
120 WAVES AND WAVE-INDUCED HYDRODYNAMICS
Figure 5.15 Effect of nonlinear friction parameter γ on swash characteristics.
Backrush – On the basis of a development similar to that used for the uprush, for
backrush V ′< 0 and Eq. (5.65) becomes
dV ′
dt ′
= −2+ γ V ′2, (5.82)
the solution of which can be shown to be
V ′(t ′) = −
√
2
γ
tanh[
√
2γ (t ′ − t ′max)] (5.83)
Equation (5.83) can be integrated to yield
x ′(t ′) = x ′max −
1
γ
�n{cosh
√
2γ (t ′ − t ′max)} (5.84)
The natural period is altered by nonlinear friction and can be determined from
Eqs. (5.79) and (5.84) as
T ′n =
1√
2γ
{
tan−1
(√
γ
2
)
+ cosh−1[eγ x ′max ]
}
(5.85)
Figure 5.15 shows the effect of varying nonlinear friction on natural period T ′n,
maximum downrush velocity (V ′0), and maximum uprush displacement x ′max.
5.6.5 FIELD AND THEORETICAL STUDIES
Hughes (1992) carried out a comparison of theoretical andmeasured swashmotions.
He applied the long-wave frictionless equations to predict the swash motions on a
planar beach face. It was found from the theory that the trajectory of the leading
edge of the swash motion was described qualitatively by Eq. (5.60) and that the
thickness of the swash decreased as the swash traversed up the beach face. The ﬁeld
experiments were conducted in southeast Australia on several beaches ranging in
P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH
CB373-05 CB373 June 27, 2001 11:28 Char Count= 0
REFERENCES 121
slope from 0.093 to 0.15. The beach faces were considerably steeper than the proﬁle
immediately seaward, leading to bore collapse on the beach face. The swash mo-
tions were documented by capacitance gauges and stakes driven into the beach face.
Documentation from the capacitance gauges was complemented by photographing
the swash motion in the vicinity of the stakes. It was found that long wave theory
provided a good representation for the form of the uprush trajectories and the swash
thicknesses; however, the magnitudes of both were overpredicted. The measured
uprush trajectories were found to be approximately 0.65 of the theoretical, and the
swash thicknesses were overpredicted by a factor of 2 to 3. It was concluded that a
major reason for the differences between measurements and theory was the lack of
friction and inﬁltration in the theoretical formulation.
REFERENCES
Abbott,M.B., H.M. Petersen, andO. Skovgaard, “On theNumericalModeling of ShortWaves
in Shallow Water,” J. Hyd. Res., 16, 173–204, 1978.
Airy, G.B., “Tides and Waves,” Encyclopaedia Metropolitana, London, J.J. Grifﬁn, 1845.
Allen, J.S., P.A. Newberger, and R.A. Holman, “Nonlinear Shear Instabilities of Alongshore
Currents on Plane Beaches,” J. Fluid Mech., 310, 181–213, 1996.
Ball, F.K., “Edge Waves in an Ocean of Finite Depth,” Deep Sea Res., Oceanogr. Abst., 14,
79–88, 1967.
Battjes, J.A., “Surf Similarity,” Proc. 14th Intl. Conf. Coastal Eng., ASCE, Copenhagen, 466–
480, 1974.
Battjes, J.A., and J.P.F.M. Janssen, “Energy Loss and Set-up due to Breaking of Random
Waves,” Proc. 16th Intl. Conf. Coastal Eng., ASCE, Hamburg, 1978.
Berkhoff, J.C.W., “Computation of Combined Refraction–Diffraction,” Proc. 13th Intl. Conf.
Coastal Eng., ASCE, Vancouver, 471–484, 1972.
Birkemeier, W.A., and R.A. Dalrymple, “Nearshore Water Circulation Induced by Wind and
Waves,” Proceedings of Modeling 75, ASCE, San Francisco, 1975.
Booij, N., “GravityWaves onWater withNonuniformDepths andCurrent,” Tech. Univ. Delft,
Rpt. 81-1, Dept. Civil Eng., 1981.
Booij, N., and L.H. Holthuijsen, “Propagation of Ocean Waves in Discrete Spectral Wave
Models,” J. Computational Physics, 68, 307–326, 1987.
Booij, N., L.H. Holthuijsen, and T.H.C. Herbers, “A Numerical Model for Wave Boundary
Conditions in Port Design,” Proc. Intl. Conf. Numerical and Physical Modeling of Ports and
Harbours,” BHRA, Birmingham, 263–268, 1985.
Bowen, A.J., “The Generation of Longshore Currents on a Plane Beach,” J. Marine Res., 37,
206–215, 1969a.
Bowen, A.J., “Rip Currents, I. Theoretical Investigations,” J. Geophys. Res., 74, 5467–5478,
1969b.
Bowen,A.J., andR.T.Guza, “EdgeWaves and Surf Beat,” J. Geophys. Res., 83, C4, 1913–1920,
1978.
Bowen, A.J., and R.A. Holman, “Shear Instabilities of the Longshore Current: 1. Theory,”
J. Geophys. Res., 94, 18023–18030, 1989.
Bowen, A.J., D.L. Inman, andV.P. Simmons, “Wave ‘Set-down’ andWave Set-up,” J. Geophys.
Res., 73, 2569–2577, 1968.
Brink-Kjaer, O., “Depth-Current Refraction of Wave Spectra,” Symp. Description and Mod-
eling of Directional Seas,” Tech. Univ. Denmark, 1984.
Carrier, G.F., and H.P. Greenspan, “Water Waves of Finite Amplitude on a Sloping Beach,”
J. Fluid Mech., 4, 97–109, 1957.
Chen, Q., R.A. Dalrymple, J.T. Kirby, A. Kennedy, and M.C. Haller, “Boussinesq Modelling
of a Rip Current System,” J. Geophys. Res., 104, C9, 20,617–20,638, 1999.
P1: FCH/SPH P2: FCH/SPH QC: FCH/SBA T1: FCH
CB373-05 CB373 June 27, 2001 11:28 Char Count= 0
122 WAVES AND WAVE-INDUCED HYDRODYNAMICS
Dally,W.R., “RandomBreakingWaves: AClosed-form Solution for Planar Beaches,”Coastal
Eng., 14, 3, 233–265, 1990.
Dally, W.R., R.G. Dean, and R.A. Dalrymple, “Wave Height Variation Across Beaches of
Arbitrary Proﬁle,” J. Geophys. Res., Vol. 90, 6, 11917–11927, 1985.
Dalrymple, R.A., “A Finite Amplitude Wave on a Linear Shear Current,” J. Geophys. Res.,
79, 30, 4498–4505, 1974.
Dalrymple, R.A., “AMechanism for Rip Current Generation on an Open Coast,” J. Geophys.
Res., 80, 3485–3487, 1975.
Dalrymple, R.A., “RipCurrents and Their Causes,”Proc. 16th Intl. Conf. Coastal Eng., ASCE,
Hamburg, 1414–1427, 1978.
Dalrymple, R.A. “A Model for the Refraction of Water Waves,” J. Waterway, Port, Coastal
and Ocean Eng., ASCE, 114, 4, 423–435, 1988.
Dalrymple, R.A., and C.J. Lozano, “Wave–Current Interaction Models for Rip Currents,”
J. Geophys. Res., Vol. 83, No. C12, 1978.
Dalrymple, R.A., J.T. Kirby, and D.J. Seli, “Wave Trapping by Breakwaters,” Proc. 20th Intl.
Conf. Coastal Eng., Taipei, ASCE, 1986.
Dean, R.G., “Stream Function Representation of Nonlinear Ocean Waves, J. Geophys. Res.,
70, 18, 4561–4572, 1965.
Dean, R.G., and R.A. Dalrymple, Water Wave Mechanics for Engineers and Scientists,
Singapore: World Scientiﬁc Press, 353 pp., 1991.
Dingemans, M.W.,Water Wave Propagation Over Uneven Bottoms, Vols. 1 and 2, Singapore:
World Scientiﬁc Press, 967 pp., 1997.
Dodd, N., J. Oltman-Shay, and E.B. Thornton, “Shear Instabilities in the Longshore Current:
A Comparison of Observations and Theory,” J. Phys Oceanogr., 22, 62–82, 1992.
Ebersole, B.A., and R.A. Dalrymple, “Numerical Modelling of Nearshore Circulation,” Proc.
17th Intl. Conf.```