Rotational waves generated by current-topography interaction. (English) Zbl 1418.35303
Summary: We study nonlinear free-surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg – de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 35R35 | Free boundary problems for PDEs |
| 76U05 | General theory of rotating fluids |
| 76B07 | Free-surface potential flows for incompressible inviscid fluids |
| 35B35 | Stability in context of PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
References:
| [1] | Teles da SilvaAF, PeregrineDH. Steep steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 1988;195:281-302. |
| [2] | VasanV, OliverasK. Pressure beneath a traveling wave with vorticity constant. Discrete Continuous Dyn Syst A. 2014;34:3219-3239. · Zbl 1292.35222 |
| [3] | RibeiroRJr., MilewskiPA, NachbinA. Flow structure beneath rotational water waves. J Fluid Mech. 2017;812:792-814. · Zbl 1383.76201 |
| [4] | ConstantinA. Nonlinear water waves: introduction and overview. Philos Trans R Soc A. 2018;376:20170310. · Zbl 1404.76004 |
| [5] | ConstantinA, StraussW, VarvarucaE. Global bifurcation of steady gravity water waves with critical layers. Acta Math. 2016;217:195-262. · Zbl 1375.35294 |
| [6] | ViottiC, DutykhD, DiasF. The conformal‐mapping method for surface gravity wave in the presence of variable bathymetry and mean current. Procedia IUTAM2014;11:110-118. |
| [7] | ViottiC, DiasF. Extreme waves induced by strong depth transitions: fully nonlinear results. Phys Fluids. 2014;26:051705. · Zbl 1321.76013 |
| [8] | AkylasTR. On the excitation of long nonlinear water waves by a moving pressure distribution. J Fluid Mech. 1984;141:455-466. · Zbl 0551.76018 |
| [9] | GrimshawRH, SmythN. Resonant flow of a stratified fluid over topography. J Fluid Mech. 1986;169:429-464. · Zbl 0614.76108 |
| [10] | WuDM, WuTY. Three‐dimensional nonlinear long waves due to moving surface pressure. In: Proc. 14th. Symp. on Naval Hydrodynamics, Nat. Acad. Sci., Washington, DC, 1982;103-25. |
| [11] | WuTY. Generation of upstream advancing solitons by moving disturbances. J Fluid Mech. 1987;184:75-99. · Zbl 0644.76017 |
| [12] | CamassaR, WuTY. Stability of forced steady solitary waves. Philos Trans R Soc Lond. 1991;A337:429-466. · Zbl 0748.76058 |
| [13] | CamassaR, WuTY. Stability of some stationary solutions for the forced KdV equation. Physica. 1991;D51:295-307. · Zbl 0742.35053 |
| [14] | ChardardF, DiasF, NguyenHY, Vanden‐BroeckJM. Stability of some stationary solutions to the forced KdV equation with one or two bumps. J Eng Math. 2011;70:175-189. · Zbl 1254.76020 |
| [15] | AlbalwiMD, MarchantTR, SmythNF. Higher‐order modulation theory for resonant flow over topography. Phys Fluids. 2017;29:77-101. |
| [16] | Vanden‐BroeckJM, TuckE. Waveless free‐surface pressure distributions. J Ship Res. 1985;29:151-158. |
| [17] | AsavanantJ, MaleewongM, ChoiJ. Computation of free‐surface flows due to pressure distribution. Korean Math Soc. 2001;16:137-152. · Zbl 1101.76373 |
| [18] | BinderBJ, DiasF, Vanden‐BroeckJM. Forced solitary waves and fronts past submerged obstacles. Chaos. 2005;15:037106. · Zbl 1144.37321 |
| [19] | GrimshawR, MaleewongM. Stability of steady gravity waves generated by a moving localized pressure disturbance in water of finite depth. Phys Fluids. 2013;25:076605. · Zbl 1320.76049 |
| [20] | ChoiW, CamassaR. Exact evolution equations for surface waves. J Eng. Mech. 1999;125:756-760. |
| [21] | DyachenkoAL, ZakharovVE, KuznetsovEA. Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys Rep. 1996;22:916-928. |
| [22] | ChoiW. Nonlinear surface waves interacting with a linear shear current. Math Comput Simul. 2009;80:29-36. · Zbl 1422.76026 |
| [23] | MilewskiP, Vanden‐BroeckJM, WangZ. Dynamics of steep two‐dimensional gravity‐capillary solitary waves. J Fluid Mech. 2010;664:466-477. · Zbl 1221.76052 |
| [24] | NachbinA. A terrain‐following Boussinesq system. SIAM J Appl Math. 2003;63:905-922. · Zbl 1078.76015 |
| [25] | BainesPG. Topographic Effects in Stratified Flows. Cambridge: Cambridge University Press; 1995. · Zbl 0840.76001 |
| [26] | GrimshawRH, YiZ. Resonant generation of finite‐amplitude waves by the flow of a uniformly stratified fluid over topography. J Fluid Mech. 1991;229:603-628. · Zbl 0850.76808 |
| [27] | WangB, RedekoppLG. Long internal waves in shear flows: topographic resonance and wave‐induced global instability. Dyn Atmos Oceans. 2001;33:263-302. |
| [28] | HelfrichKR, MelvilleWK. Long nonlinear internal waves. Annu Rev Fluid Mech. 2006;38:395-425. · Zbl 1098.76018 |
| [29] | LambKG. Internal wave breaking dissipation mechanism on the continental slope/shelf. Annu Rev Fluid Mech. 2014;46:231-254. · Zbl 1297.76043 |
| [30] | SoontiensN, SubichC, StastnaM. Numerical simulation of supercritical trapped internal waves over topography. Phys Fluids. 2010;22:116605. · Zbl 1308.76056 |
| [31] | StastnaM, PeltierWR. On the resonant generation of large‐amplitude internal solitary‐like waves. J Fluid Mech. 2005;543:267-292. · Zbl 1137.76328 |
| [32] | WheelerMH. The Froude number for solitary water waves with vorticity. J Fluid Mech. 2015;768:91-112. · Zbl 1337.76011 |
| [33] | WhithamGB. Linear and Nonlinear Waves. New York: Wiley; 1974. · Zbl 0373.76001 |
| [34] | FreemanNC, JohnsonRS. Shallow water waves on shear flows. J Fluid Mech. 1970;42:401-409. · Zbl 0202.27002 |
| [35] | JohnsonRS. A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge: Cambridge University Press; 1997. · Zbl 0892.76001 |
| [36] | TrefethenLN. Spectral Methods in MATLAB. Philadelphia: SIAM; 2001. |
| [37] | MilewskiP, TabakEG. A pseudo‐spectral algorithm for the solution of nonlinear wave equations. SIAM J Sci Comp. 1999;21:1102-1114. · Zbl 0953.65073 |
| [38] | Vanden‐BroeckJM. Steep solitary waves in water of finite depth with constant vorticity. J Fluid Mech. 1994;274:339-348. · Zbl 0814.76027 |
| [39] | BoydJ. New directions in solitons and nonlinear periodic waves. In: HutchinsonJW (ed.), WuTY (ed.), eds. Advances in Applied Mechanics. New York: Academic Press; 1990. |
| [40] | BoydJ. Weakly non‐local solitons for capillary‐gravity waves: fifth‐degree Korteweg-de‐Vries equation. Phys D. 1990;48:129-146. · Zbl 0728.35100 |
| [41] | AkylasTR, GrimshawR. Solitary internal waves with oscillatory tails. J Fluid Mech. 1992;242:279-298. · Zbl 0754.76014 |
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