A two-layer viscous Boussinesq-type model is developed to simulate the wave energy dissipation during wave propagation in deep water. The viscous terms are incorporated into both the dynamic and kinematic boundary conditions at the free surface, and the corresponding analytical solution of the second-order amplitude has been derived for the first time. The linear and nonlinear properties of the model are analyzed with different viscosity coefficients. When the viscosity coefficient is 1 × 10−4 m2/s, the linear phase velocity, decay rate, second-order amplitude, and velocity profiles of the viscous model are accurate for up to h/L0 (h is water depth, L0 is characteristic wavelength) ≈ 8.66, 5.86, 3.60, 3.60, and 7.51 within 1% error, respectively. The finite difference method is adopted for the numerical implementation of the model. To verify the linear and nonlinear properties of the model, computed results for linear waves and focused wave group in deep water are compared with linear analytical solutions and experimental data, respectively.
References
1.
N. J.
Zabusky
and C. J.
Galvin
, “Shallow-water waves, the Korteweg–deVries equation and soliton
,” J. Fluid Mech.
47
(4
), 811
–824
(1971
).2.
J. L.
Bona
, W. G.
Pritchard
, and L. R.
Scott
, “An evaluation of a model equation for water waves
,” Philos. Trans. R. Soc. London, Ser. A
302
(1471
), 457
–510
(1981
).3.
F.
Ardhuin
, B.
Chapron
, and F.
Collard
, “Observation of swell dissipation across oceans
,” Geophys. Res. Lett.
36
(6
), L06607
, (2009
).4.
D. M.
Henderson
and H.
Segur
, “The role of dissipation in the evolution of ocean swell
,” J. Geophys. Res. Oceans
118
(10
), 5074
–5091
, (2013
).5.
C. R.
Zaug
and J. D.
Carter
, “Dissipative models of swell propagation across the Pacific
,” Stud. Appl. Math.
147
(4
), 1519
–1537
(2021
).6.
H.
Segur
, D.
Henderson
, J.
Carter
, J.
Hammack
, C. M.
Li
, D.
Pheiff
, and K.
Socha
, “Stabilizing the Benjamin–Feir instability
,” J. Fluid Mech.
539
, 229
–271
(2005
).7.
G.
Wu
, Y.
Liu
, and D. K.
Yue
, “A note on stabilizing the Benjamin–Feir instability
,” J. Fluid Mech.
556
, 45
–54
(2006
).8.
Y.
Ma
, G.
Dong
, M.
Perlin
, X.
Ma
, and G.
Wang
, “Experimental investigation on the evolution of the modulation instability with dissipation
,” J. Fluid Mech.
711
, 101
–121
(2012
).9.
A.
Armaroli
, D.
Eeltink
, M.
Brunetti
, and J.
Kasparian
, “Nonlinear stage of Benjamin–Feir instability in forced/damped deep-water waves
,” Phys. Fluids
30
, 017102
(2018
).10.
J. D.
Carter
and A.
Govan
, “Frequency downshift in a viscous fluid
,” Eur. J. Mech. B
59
, 177
–185
(2016
).11.
J. D.
Carter
, D.
Henderson
, and I.
Butterfield
, “A comparison of frequency downshift models of wave trains on deep water
,” Phys. Fluids
31
(1
), 013103
(2019
).12.
B.
Molin
, “On the piston and sloshing modes in moonpools
,” J. Fluid Mech.
430
, 27
–50
(2001
).13.
Y.
Liu
and H. J.
Li
, “A new semi-analytical solution for gap resonance between twin rectangular boxes
,” Proc. Inst. Mech. Eng., Part M
228
(1
), 3
–16
(2014
).14.
Y.
Liu
, H. J.
Li
, L.
Lu
, A. J.
Li
, and L.
Tan
, “A semi-analytical potential solution for wave resonance in gap between floating box and vertical wall
,” China Ocean Eng.
34
(6
), 747
–759
(2020
).15.
Z.
Tian
, M.
Perlin
, and W.
Choi
, “Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model
,” J. Fluid Mech.
655
, 217
–257
(2010
).16.
Z.
Tian
, M.
Perlin
, and W.
Choi
, “An eddy viscosity model for two-dimensional breaking waves and its validation with laboratory experiments
,” Phys. Fluids
24
(3
), 036601
(2012
).17.
N.
Pizzo
, E.
Murray
, D. L.
Smith
, and L.
Lenain
, “The role of bandwidth in setting the breaking slope threshold of deep-water focusing wave packets
,” Phys. Fluids
33
, 111706
(2021
).18.
Y.
Liu
, W.
Cao
, W.
Zhang
, and Z.
Xia
, “Analysis on numerical stability and convergence of Reynolds averaged Navier–Stokes simulations from the perspective of coupling modes
,” Phys. Fluids
34
, 015120
(2022
).19.
J.
Boussinesq
, “Lois de l'extinction de la houle en haute mer
,” C. R. Acad. Sci. Paris
121
(15–20
), 2
(1895
).20.
21.
K. D.
Ruvinsky
and G. I.
Freidman
, “The fine structure of strong gravity-capillary waves
,” Nonlinear Waves: Structures and Bifurcations
, edited by A. V.
Gaponov-Grekhov
and M. I.
Rabinovich
(Nauka
, 1987
), pp. 304
–326
.22.
K. D.
Ruvinsky
, F. I.
Feldstein
, and G. I.
Freidman
, “Numerical simulations of the quasi-stationary stage of ripple excitation by steep gravity–capillary waves
,” J. Fluid Mech.
230
, 339
–353
(1991
).23.
M. S.
Longuet-Higgins
, “Theory of weakly damped Stokes waves: A new formulation and its physical interpretation
,” J. Fluid Mech.
235
, 319
–324
(1992
).24.
L.
Jiang
, C. L.
Ting
, M.
Perlin
, and W. W.
Schultz
, “Moderate and steep Faraday waves: Instabilities, modulation and temporal asymmetries
,” J. Fluid Mech.
329
, 275
–307
(1996
).25.
D. D.
Joseph
and J.
Wang
, “The dissipation approximation and viscous potential flow
,” J. Fluid Mech.
505
, 365
–377
(2004
).26.
J.
Wang
and D. D.
Joseph
, “Purely irrotational theories of the effect of the viscosity on the decay of free gravity waves
,” J. Fluid Mech.
559
, 461
–472
(2006
).27.
F.
Dias
, A. I.
Dyachenko
, and V. E.
Zakharov
, “Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions
,” Phys. Lett. A
372
(8
), 1297
–1302
(2008
).28.
D.
Dutykh
and F.
Dias
, “Viscous potential free-surface flows in a fluid layer of finite depth
,” C. R. Math.
345
(2
), 113
–118
(2007
).29.
D.
Dutykh
and F.
Dias
, “Dissipative Boussinesq equations
,” C. R. Mec.
335
(9–10
), 559
–583
(2007
).30.
D.
Dutykh
, “Visco-potential free-surface flows and long wave modelling
,” Eur. J. Mech. B
28
(3
), 430
–443
(2009
).31.
M.
Kakleas
and D. P.
Nicholls
, “Numerical simulation of a weakly nonlinear model for water waves with viscosity
,” J. Sci. Comput.
42
(2
), 274
–290
(2010
).32.
R.
Granero-Belinchón
and S.
Scrobogna
, “Models for damped water waves
,” SIAM J. Appl. Math.
79
(6
), 2530
–2550
(2019
).33.
L. F.
Mouassom
, T. N.
Nkomom
, A.
Mvogo
, and C. B.
Mbane
, “Effects of viscosity and surface tension on soliton dynamics in the generalized KdV equation for shallow water waves
,” Commun. Nonlinear Sci. Numer. Simul.
102
, 105942
(2021
).34.
S.
Li
, S.
Qian
, H.
Chen
, J.
Song
, and A.
Cao
, “An extended nonlinear Schrödinger equation for water waves with linear shear flow, wind, and dissipation
,” AIP Adv.
11
(2
), 025326
(2021
).35.
B.
Liao
, G.
Dong
, Y.
Ma
, X.
Ma
, and M.
Perlin
, “Modified nonlinear Schrödinger equation for gravity waves with influence of wind, currents and dissipation
,” Phys. Fluids
35
, 037103
(2023
).36.
D. M.
Ambrose
, J. L.
Bona
, and D. P.
Nicholls
, “Well-posedness of a model for water waves with viscosity
,” Discrete Contin. Dyn. Syst. B
17
(4
), 1113
(2012
).37.
M.
Ngom
and D. P.
Nicholls
, “Well-posedness and analyticity of solutions to a water wave problem with viscosity
,” J. Differ. Equations
265
(10
), 5031
–5065
(2018
).38.
R.
Granero-Belinchón
and S.
Scrobogna
, “Well-posedness of the water-wave with viscosity problem
,” J. Differ. Equations
276
, 96
–148
(2021
).39.
J. M.
Witting
, “A unified model for the evolution nonlinear water waves
,” J. Comput. Phys.
56
(2
), 203
–236
(1984
).40.
O.
Nwogu
, “Alternative form of Boussinesq equations for nearshore wave propagation
,” J. Waterw., Port, Coastal, Ocean Eng.
119
(6
), 618
–638
(1993
).41.
G.
Wei
, J. T.
Kirby
, S. T.
Grilli
, and R.
Subramanya
, “A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear unsteady waves
,” J. Fluid Mech.
294
, 71
–92
(1995
).42.
S.
Beji
and K.
Nadaoka
, “A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth
,” Ocean Eng.
23
(8
), 691
–704
(1996
).43.
P. A.
Madsen
and H. A.
Schäffer
, “Higher-order Boussinesq–type equations for surface gravity waves: Derivation and analysis
,” Philos. Trans. R. Soc. London, Ser. A
356
(1749
), 3123
–3181
(1998
).44.
M. F.
Gobbi
, J. T.
Kirby
, and G. E.
Wei
, “A fully nonlinear Boussinesq model for surface waves. II. Extension to O(kh)4
,” J. Fluid Mech.
405
, 181
–210
(2000
).45.
P.
Lynett
and P. L. F.
Liu
, “A two-layer approach to wave modelling
,” Proc. R Soc. London, A
460
(2049
), 2637
–2669
(2004
).46.
Y.
Agnon
, P. A.
Madsen
, and H. A.
Schäffer
, “A new approach to high-order Boussinesq models
,” J. Fluid Mech.
399
, 319
–333
(1999
).47.
P. A.
Madsen
, H. B.
Bingham
, and H.
Liu
, “A new Boussinesq method for fully nonlinear waves from shallow to deep water
,” J. Fluid Mech.
462
, 1
–30
(2002
).48.
F.
Chazel
, M.
Benoit
, A.
Ern
, and S.
Piperno
, “A double-layer Boussinesq-type model for highly nonlinear and dispersive waves
,” Proc. R. Soc. A
465
(2108
), 2319
–2346
(2009
).49.
P. A.
Madsen
, H. B.
Bingham
, and H. A.
Schäffer
, “Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis
,” Proc. R. Soc. London, A
459
(2033
), 1075
–1104
(2003
).50.
Z. B.
Liu
and K. Z.
Fang
, “A new two-layer Boussinesq model for coastal waves from deep to shallow water: Derivation and analysis
,” Wave Motion
67
, 1
–14
(2016
).51.
Z. B.
Liu
and K. Z.
Fang
, “Numerical verification of a two-layer Boussinesq-type model for surface gravity wave evolution
,” Wave Motion
85
, 98
–113
(2019
).52.
K. Z.
Fang
, Z. B.
Liu
, J. W.
Sun
, Z. H.
Xie
, and Z. Y.
Zheng
, “Development and validation of a two-layer Boussinesq model for simulating free surface waves generated by bottom motion
,” Appl. Ocean Res.
94
, 101977
(2020
).53.
K. Z.
Fang
, Z. B.
Liu
, P.
Wang
, H.
Wu
, J. W.
Sun
, and J.
Yin
, “Modeling solitary wave propagation and transformation over complex bathymetries using a two-layer Boussinesq model
,” Ocean Eng.
265
, 112549
(2022
).54.
P. L. F.
Liu
and A.
Orfila
, “Viscous effects on transient long-wave propagation
,” J. Fluid Mech.
520
, 83
–92
(2004
).55.
P. J.
Lynett
, T. R.
Wu
, and P. L. F.
Liu
, “Modeling wave runup with depth-integrated equations
,” Coastal Eng.
46
(2
), 89
–107
(2002
).56.
P. L. F.
Liu
, G.
Simarro
, J.
Vandever
, and A.
Orfila
, “Experimental and numerical investigation of viscous effects on solitary wave propagation in a wave tank
,” Coastal Eng.
53
(2–3
), 181
–190
(2006
).57.
A.
Orfila
, G.
Simarro
, and P. L. F.
Liu
, “Bottom friction and its effects on periodic long wave propagation
,” Coastal Eng.
54
(11
), 856
–864
(2007
).58.
H. V.
Le Meur
, “Derivation of a viscous Boussinesq system for surface water waves
,” Asymptotic Anal.
94
(3–4
), 309
–345
(2015
).59.
D.
Dutykh
and O.
Goubet
, “Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach
,” Math. Comput. Simul.
127
, 80
–93
(2016
).60.
J. N.
Hunt
, “The viscous damping of gravity waves in shallow water
,” La Houille Blanche
50
(6
), 685
–691
(1964
).61.
D. E.
Horsley
and L. K.
Forbes
, “A spectral method for Faraday waves in rectangular tanks
,” J. Eng. Math.
79
(1
), 13
–33
(2013
).62.
H.
Liang
and X.
Chen
, “Viscous effects on the fundamental solution to ship waves
,” J. Fluid Mech.
879
, 744
–774
(2019
).63.
J. T.
Kirby
, G.
Wei
, Q.
Chen
, A. B.
Kennedy
, and R. A.
Dalrymple
, “FUNWAVE 1.0: Fully nonlinear Boussinesq wave model—Documentation and user's manual
,” Report No. CACR-98-06, 1998
.64.
Z. B.
Liu
, K. Z.
Fang
, and Y. Z.
Cheng
, “A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed
,” J. Fluid Mech.
842
, 323
–353
(2018
).65.
T. E.
Baldock
, C.
Swan
, and P. H.
Taylor
, “A laboratory study of nonlinear surface waves on water
,” Philos. Trans. R. Soc. London, Ser. A
354
(1707
), 649
–676
(1996
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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