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Water Waves
Number of Followers: 0 Hybrid journal (It can contain Open Access articles) ISSN (Print) 2523-367X - ISSN (Online) 2523-3688 Published by Springer-Verlag [2653 journals] |
- A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave
Interaction with Fixed Structures- Abstract: We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global \(L^2\) projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.
PubDate: 2019-10-11
DOI: 10.1007/s42286-019-00018-5
- Abstract: We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global \(L^2\) projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.
- Phase Dynamics of the Dysthe Equation and the Bifurcation of Plane Waves
- Abstract: The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such occurrences may be described using a Korteweg–de Vries equation. The solitary wave solutions of this system form a qualitative prototype for the bifurcating dynamics, and the role of mean flow and steepening is then made clear through how they enter the amplitude and width of these solitary waves. In addition, higher order phase dynamics are investigated, leading to increased nonlinear regimes which in turn have a more profound impact on how the plane waves transform under defects in the phase.
PubDate: 2019-10-08
DOI: 10.1007/s42286-019-00016-7
- Abstract: The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such occurrences may be described using a Korteweg–de Vries equation. The solitary wave solutions of this system form a qualitative prototype for the bifurcating dynamics, and the role of mean flow and steepening is then made clear through how they enter the amplitude and width of these solitary waves. In addition, higher order phase dynamics are investigated, leading to increased nonlinear regimes which in turn have a more profound impact on how the plane waves transform under defects in the phase.
- Fully Nonlinear Potential Flow Simulations of Wave Shoaling Over Slopes:
Spilling Breaker Model and Integral Wave Properties- Abstract: A spilling breaker model (SBM) is implemented in an existing two-dimensional (2D) numerical wave tank (NWT), based on fully nonlinear potential flow (FNPF) theory and the boundary element method (BEM), in which an absorbing surface pressure is specified over the crest of impending breaking waves (detected based on a maximum front slope criterion) to simulate the power dissipated during breaking. The latter is calibrated to match that of a hydraulic jump of parameters identical to local wave properties (height, celerity, \(\ldots \) ). Although this model is not aimed at being fully physical in the surfzone, it allows more accurately simulating properties of fully nonlinear shoaling waves than standard empirical absorbing beaches (AB). After assessing the convergence with the discretization of 2D-NWT results for strongly nonlinear shoaling waves, simulations are first validated based on laboratory experiments for non-breaking and breaking waves, shoaling up a mild slope; a good agreement is found between both. The NWT is then used to compute fully nonlinear local (height, celerity, asymmetry) and integral (mean-water-level, radiation stress) properties of periodic waves shoaling over mild slopes. Discrepancies with standard results of wave theories are discussed in light of fully nonlinear effects modeled in the NWT. While only periodic waves are considered here, the SBM could be applied to shoaling irregular waves, for which the breaking point location will constantly vary, which would be advantageous compared to an AB fixed in space. For such cases, besides its more physical energy absorption, the SBM would allow better preventing wave overturning that may occur far from shore due to nonlinear wave–wave interactions and otherwise interrupt the FNPF–NWT simulations.
PubDate: 2019-10-08
DOI: 10.1007/s42286-019-00017-6
- Abstract: A spilling breaker model (SBM) is implemented in an existing two-dimensional (2D) numerical wave tank (NWT), based on fully nonlinear potential flow (FNPF) theory and the boundary element method (BEM), in which an absorbing surface pressure is specified over the crest of impending breaking waves (detected based on a maximum front slope criterion) to simulate the power dissipated during breaking. The latter is calibrated to match that of a hydraulic jump of parameters identical to local wave properties (height, celerity, \(\ldots \) ). Although this model is not aimed at being fully physical in the surfzone, it allows more accurately simulating properties of fully nonlinear shoaling waves than standard empirical absorbing beaches (AB). After assessing the convergence with the discretization of 2D-NWT results for strongly nonlinear shoaling waves, simulations are first validated based on laboratory experiments for non-breaking and breaking waves, shoaling up a mild slope; a good agreement is found between both. The NWT is then used to compute fully nonlinear local (height, celerity, asymmetry) and integral (mean-water-level, radiation stress) properties of periodic waves shoaling over mild slopes. Discrepancies with standard results of wave theories are discussed in light of fully nonlinear effects modeled in the NWT. While only periodic waves are considered here, the SBM could be applied to shoaling irregular waves, for which the breaking point location will constantly vary, which would be advantageous compared to an AB fixed in space. For such cases, besides its more physical energy absorption, the SBM would allow better preventing wave overturning that may occur far from shore due to nonlinear wave–wave interactions and otherwise interrupt the FNPF–NWT simulations.
- Experimental Realization of Periodic Deep-Water Wave Envelopes with and
without Dissipation- Abstract: The Korteweg–de Vries equation that describes surface gravity water wave dynamics in shallow water is well known to admit cnoidal wave solutions, i.e. periodic travelling waves with stationary wave shape. Such type of periodic wave patterns can be also found in the deep-water waves with the envelopes that follow the dynamics of the nonlinear Schrödinger equation (NLS). A particular class of NLS periodic and stationary solutions are cnoidal (CN) and dnoidal (DN) envelopes. This one parameter family of solutions has as limiting cases either the envelope soliton or a constant background. In this experimental study, we discuss the physical features of such waves and emphasize the particular effect of dissipation, as observed in several hydrodynamic experiments that have been conducted in several water wave facilities with different dimensions. Experiments on such type of periodic wave envelopes in a large facility have been already reported in the early 1990s. These studies demonstrated significant deviation of the DN-type envelopes from theory. Here, we show that these deviations are due to the effect of dissipation that can be qualitatively considered by adapting the NLS framework accordingly. Reducing the amplitude of the carrier wave makes the wave field susceptible to dissipation effects. Our experiments prove that the dissipation is indeed responsible for phase-shift pulsations for DN-type envelopes.
PubDate: 2019-09-19
DOI: 10.1007/s42286-019-00015-8
- Abstract: The Korteweg–de Vries equation that describes surface gravity water wave dynamics in shallow water is well known to admit cnoidal wave solutions, i.e. periodic travelling waves with stationary wave shape. Such type of periodic wave patterns can be also found in the deep-water waves with the envelopes that follow the dynamics of the nonlinear Schrödinger equation (NLS). A particular class of NLS periodic and stationary solutions are cnoidal (CN) and dnoidal (DN) envelopes. This one parameter family of solutions has as limiting cases either the envelope soliton or a constant background. In this experimental study, we discuss the physical features of such waves and emphasize the particular effect of dissipation, as observed in several hydrodynamic experiments that have been conducted in several water wave facilities with different dimensions. Experiments on such type of periodic wave envelopes in a large facility have been already reported in the early 1990s. These studies demonstrated significant deviation of the DN-type envelopes from theory. Here, we show that these deviations are due to the effect of dissipation that can be qualitatively considered by adapting the NLS framework accordingly. Reducing the amplitude of the carrier wave makes the wave field susceptible to dissipation effects. Our experiments prove that the dissipation is indeed responsible for phase-shift pulsations for DN-type envelopes.
- Wave Kinematics in a Two-Dimensional Plunging Breaker
- Abstract: In the wake of theoretical, numerical and experimental advances by a large number of contributors, we revisit here some aspects of the fluid kinematics in a two-dimensional plunging breaker occurring in shallow water. In particular, we propose a simplified identification of the velocity distribution at the free surface in terms of the velocity at some characteristic points. We can then simply explain the reasons for which the velocity is maximum inside the barrel at its roof. We also show that the relative velocity field calculated in a coordinate system centered to a point where the velocity is maximum may have a possible analytic representation.
PubDate: 2019-08-08
DOI: 10.1007/s42286-019-00013-w
- Abstract: In the wake of theoretical, numerical and experimental advances by a large number of contributors, we revisit here some aspects of the fluid kinematics in a two-dimensional plunging breaker occurring in shallow water. In particular, we propose a simplified identification of the velocity distribution at the free surface in terms of the velocity at some characteristic points. We can then simply explain the reasons for which the velocity is maximum inside the barrel at its roof. We also show that the relative velocity field calculated in a coordinate system centered to a point where the velocity is maximum may have a possible analytic representation.
- Account of Occasional Wave Breaking in Numerical Simulations of Irregular
Water Waves in the Focus of the Rogue Wave Problem- Abstract: The issue of accounting of the wave breaking phenomenon in direct numerical simulations of oceanic waves is discussed. It is emphasized that this problem is crucial for the deterministic description of waves, and also for the dynamical calculation of extreme wave statistical characteristics, such as rogue wave height probability, asymmetry, etc. The conditions for accurate simulations of irregular steep waves within the High Order Spectral Method for the potential Euler equations are identified. Such non-dissipative simulations are considered as the reference when comparing with the simulations of occasionally breaking waves which use two kinds of wave breaking regularization. It is shown that the perturbations caused by the wave breaking attenuation may be noticeable within 20 min of the performed simulation of the wave evolution.
PubDate: 2019-08-08
DOI: 10.1007/s42286-019-00014-9
- Abstract: The issue of accounting of the wave breaking phenomenon in direct numerical simulations of oceanic waves is discussed. It is emphasized that this problem is crucial for the deterministic description of waves, and also for the dynamical calculation of extreme wave statistical characteristics, such as rogue wave height probability, asymmetry, etc. The conditions for accurate simulations of irregular steep waves within the High Order Spectral Method for the potential Euler equations are identified. Such non-dissipative simulations are considered as the reference when comparing with the simulations of occasionally breaking waves which use two kinds of wave breaking regularization. It is shown that the perturbations caused by the wave breaking attenuation may be noticeable within 20 min of the performed simulation of the wave evolution.
- The Error in Predicted Phase Velocity of Surface Waves atop a Shear
Current with Uncertainty- Abstract: The effect of a depth-dependent shear current U(z) on surface wave dispersion is conventionally calculated by assuming U(z) to be an exactly known function, from which the resulting phase velocity c(k) is determined. This, however, is not the situation in reality. Field measurements of the current profile are performed at a finite number of discrete depths and with nonzero experimental uncertainty. Here we analyse how imperfect knowledge of U(z) affects estimates of c(k). We performed a numerical experiment simulating a large number of “measurements” of three different shear currents: an exponential profile, a 1 / 7-law profile, and a profile measured in the Columbia River delta. A number of measurement points were specified, the topmost of which at \(z=-h_s\) (permitting simulation of measurement points which do not fully extend to the surface at \(z = 0\) ), and measurements taken from a normal distribution with standard deviation \(\varDelta U\) . Four different methods of reconstructing a continuous U(z) from the measurements are compared with respect to mean value and variance of c(k). We find that an ordinary least-squares polynomial fit seems robust against mispredicting mean values at the expense of relatively high variance. Its performance is similar for all profiles, whereas a fit to an exponential form is excellent in one case and poor in another. A clear conclusion is the need for a measurement of the surface velocity U(0) when there is significant shear near the surface. For the exponential and Columbia profiles alike, errors due to extrapolation of U from \(z=-h_s\) to 0 dominate the resulting error of c, especially for shorter wavelengths. In contrast, the error in c(k) decreases slowly with a higher density of measurement points, indicating that better, not more, velocity measurements should be invested in. A pseudospectral analysis of the linear operator corresponding to the three velocity profiles was performed. In all cases, the pseudospectrum shows strong asymmetry around the eigenvalue for c, indicating that a perturbation in the underlying current is more likely to push c to higher, not lower, values. This is in tentative agreement with our observation that for sufficiently large \(\varDelta U\) , c is found to have predominantly positive skewness, although the direct relationship between the two is not altogether obvious.
PubDate: 2019-07-18
DOI: 10.1007/s42286-019-00012-x
- Abstract: The effect of a depth-dependent shear current U(z) on surface wave dispersion is conventionally calculated by assuming U(z) to be an exactly known function, from which the resulting phase velocity c(k) is determined. This, however, is not the situation in reality. Field measurements of the current profile are performed at a finite number of discrete depths and with nonzero experimental uncertainty. Here we analyse how imperfect knowledge of U(z) affects estimates of c(k). We performed a numerical experiment simulating a large number of “measurements” of three different shear currents: an exponential profile, a 1 / 7-law profile, and a profile measured in the Columbia River delta. A number of measurement points were specified, the topmost of which at \(z=-h_s\) (permitting simulation of measurement points which do not fully extend to the surface at \(z = 0\) ), and measurements taken from a normal distribution with standard deviation \(\varDelta U\) . Four different methods of reconstructing a continuous U(z) from the measurements are compared with respect to mean value and variance of c(k). We find that an ordinary least-squares polynomial fit seems robust against mispredicting mean values at the expense of relatively high variance. Its performance is similar for all profiles, whereas a fit to an exponential form is excellent in one case and poor in another. A clear conclusion is the need for a measurement of the surface velocity U(0) when there is significant shear near the surface. For the exponential and Columbia profiles alike, errors due to extrapolation of U from \(z=-h_s\) to 0 dominate the resulting error of c, especially for shorter wavelengths. In contrast, the error in c(k) decreases slowly with a higher density of measurement points, indicating that better, not more, velocity measurements should be invested in. A pseudospectral analysis of the linear operator corresponding to the three velocity profiles was performed. In all cases, the pseudospectrum shows strong asymmetry around the eigenvalue for c, indicating that a perturbation in the underlying current is more likely to push c to higher, not lower, values. This is in tentative agreement with our observation that for sufficiently large \(\varDelta U\) , c is found to have predominantly positive skewness, although the direct relationship between the two is not altogether obvious.
- Linear Modes for Channels of Constant Cross-Section and Approximate
Dirichlet–Neumann Operators- Abstract: We study normal modes for the linear water wave problem in infinite straight channels of bounded constant cross-section. Our goal is to compare semi-analytic normal mode solutions known in the literature for special triangular cross-sections, namely isosceles triangles of equal angle of \(45^{\circ }\) and \(60^{\circ }\) , see Lamb (Hydrodynamics. Cambridge University Press, Cambridge, 1932) , Macdonald (Proc Lond Math Soc 1:101–113, 1893), Greenhill (Am J Math 97–112, 1887), Packham (Q J Mech Appl Math 33:179–187, 1980), and Groves (Q J Mech Appl Math 47:367–404, 1994), to numerical solutions obtained using approximations of the non-local Dirichlet–Neumann operator for linear waves, specifically an ad-hoc approximation proposed in Vargas-Magaña and Panayotaros (Wave Motion 65:156–174, 2016), and a first-order truncation of the systematic depth expansion by Craig et al. (Proc R Soc Lond A: Math, Phys Eng Sci 46:839–873, 2005). We consider cases of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with sinusoidal dependence in the longitudinal direction. The triangular geometries considered have slopping beach boundaries that should in principle limit the applicability of the approximate Dirichlet–Neumann operators. We nevertheless see that the approximate operators give remarkably close results for transverse even modes, while for odd transverse modes we have some discrepancies near the boundary. In the case of longitudinal modes, where the theory only yields even modes, the different approximate operators show more discrepancies for the first two longitudinal modes and better agreement for higher modes. The ad-hoc approximation is generally closer to exact modes away from the boundary.
PubDate: 2019-06-14
DOI: 10.1007/s42286-019-00010-z
- Abstract: We study normal modes for the linear water wave problem in infinite straight channels of bounded constant cross-section. Our goal is to compare semi-analytic normal mode solutions known in the literature for special triangular cross-sections, namely isosceles triangles of equal angle of \(45^{\circ }\) and \(60^{\circ }\) , see Lamb (Hydrodynamics. Cambridge University Press, Cambridge, 1932) , Macdonald (Proc Lond Math Soc 1:101–113, 1893), Greenhill (Am J Math 97–112, 1887), Packham (Q J Mech Appl Math 33:179–187, 1980), and Groves (Q J Mech Appl Math 47:367–404, 1994), to numerical solutions obtained using approximations of the non-local Dirichlet–Neumann operator for linear waves, specifically an ad-hoc approximation proposed in Vargas-Magaña and Panayotaros (Wave Motion 65:156–174, 2016), and a first-order truncation of the systematic depth expansion by Craig et al. (Proc R Soc Lond A: Math, Phys Eng Sci 46:839–873, 2005). We consider cases of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with sinusoidal dependence in the longitudinal direction. The triangular geometries considered have slopping beach boundaries that should in principle limit the applicability of the approximate Dirichlet–Neumann operators. We nevertheless see that the approximate operators give remarkably close results for transverse even modes, while for odd transverse modes we have some discrepancies near the boundary. In the case of longitudinal modes, where the theory only yields even modes, the different approximate operators show more discrepancies for the first two longitudinal modes and better agreement for higher modes. The ad-hoc approximation is generally closer to exact modes away from the boundary.
- Numerical Simulations of Modulated Waves in a Higher-Order Dysthe Equation
- Abstract: The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrödinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.
PubDate: 2019-06-13
DOI: 10.1007/s42286-019-00011-y
- Abstract: The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrödinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.
- Particle Trajectories in Nonlinear Schrödinger Models
- Abstract: The nonlinear Schrödinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile \(\eta \) , and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential \(\phi \) can be reconstructed in a similar way as the free surface profile. This observation opens up a range of potential applications since the nonlinear Schrödinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrödinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.
PubDate: 2019-05-16
DOI: 10.1007/s42286-019-00008-7
- Abstract: The nonlinear Schrödinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile \(\eta \) , and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential \(\phi \) can be reconstructed in a similar way as the free surface profile. This observation opens up a range of potential applications since the nonlinear Schrödinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrödinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.
- Editorial
- PubDate: 2019-05-01
DOI: 10.1007/s42286-019-00009-6
- PubDate: 2019-05-01
- The Pressure Boundary Condition and the Pressure as Lagrangian for Water
Waves- Abstract: The pressure boundary condition for the full Euler equations with a free surface and general vorticity field is formulated in terms of a generalized Bernoulli equation deduced from the Gavrilyuk–Kalisch–Khorsand conservation law. The use of pressure as a Lagrangian density, as in Luke’s variational principle, is reviewed and extension to a full vortical flow is attempted with limited success. However, a new variational principle for time-dependent water waves in terms of the stream function is found. The variational principle generates vortical boundary conditions but with a harmonic stream function. Other aspects of vorticity in variational principles are also discussed.
PubDate: 2019-05-01
DOI: 10.1007/s42286-019-00001-0
- Abstract: The pressure boundary condition for the full Euler equations with a free surface and general vorticity field is formulated in terms of a generalized Bernoulli equation deduced from the Gavrilyuk–Kalisch–Khorsand conservation law. The use of pressure as a Lagrangian density, as in Luke’s variational principle, is reviewed and extension to a full vortical flow is attempted with limited success. However, a new variational principle for time-dependent water waves in terms of the stream function is found. The variational principle generates vortical boundary conditions but with a harmonic stream function. Other aspects of vorticity in variational principles are also discussed.
- Low Regularity Solutions for Gravity Water Waves
- Abstract: We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in \(H^s\) , \(s > \frac{d}{2} + 1 - \mu \) , where \(\mu = \frac{1}{10}\) in the case \(d = 1\) and \(\mu = \frac{1}{5}\) in the case \(d \ge 2\) , extending previous results of Alazard–Burq–Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.
PubDate: 2019-05-01
DOI: 10.1007/s42286-019-00002-z
- Abstract: We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in \(H^s\) , \(s > \frac{d}{2} + 1 - \mu \) , where \(\mu = \frac{1}{10}\) in the case \(d = 1\) and \(\mu = \frac{1}{5}\) in the case \(d \ge 2\) , extending previous results of Alazard–Burq–Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.
- Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth:
Derivation and Numerical Solution- Abstract: The nonlinear two-dimensional problem describing periodic steady gravity waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko’s equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the \(2 \pi \) -periodic Hilbert transform \({\mathcal {C}}\) used in the equation for deep water, the equation obtained here contains a certain operator \({\mathcal {B}}_r\) , which is the sum of \({\mathcal {C}}\) and a compact operator depending on a parameter related to the depth of water. Numerical computations are based on an equivalent form of Babenko’s equation derived by virtue of the spectral decomposition of the operator \({\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t\) . Bifurcation curves and wave profiles of the extreme form are obtained numerically.
PubDate: 2019-05-01
DOI: 10.1007/s42286-019-00007-8
- Abstract: The nonlinear two-dimensional problem describing periodic steady gravity waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko’s equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the \(2 \pi \) -periodic Hilbert transform \({\mathcal {C}}\) used in the equation for deep water, the equation obtained here contains a certain operator \({\mathcal {B}}_r\) , which is the sum of \({\mathcal {C}}\) and a compact operator depending on a parameter related to the depth of water. Numerical computations are based on an equivalent form of Babenko’s equation derived by virtue of the spectral decomposition of the operator \({\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t\) . Bifurcation curves and wave profiles of the extreme form are obtained numerically.
- Rigorous Asymptotic Models of Water Waves
- Abstract: We develop a rigorous asymptotic derivation of two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting \( \epsilon \) denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in \( \epsilon \) to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water wave system is obtained as an infinite sum of solutions to linear problems at each \(O( \epsilon ^k)\) level, and truncation of this series leads to our two asymptotic models, which we call the quadratic and cubic h-models. These models are well posed in spaces of analytic functions. We prove error bounds for the difference between solutions of the h-models and the water wave system. We also show that the Craig–Sulem models of water waves can be obtained from our asymptotic procedure. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water wave system. For three very different examples, we show that the agreement between the model equations and the water wave solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.
PubDate: 2019-05-01
DOI: 10.1007/s42286-019-00005-w
- Abstract: We develop a rigorous asymptotic derivation of two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting \( \epsilon \) denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in \( \epsilon \) to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water wave system is obtained as an infinite sum of solutions to linear problems at each \(O( \epsilon ^k)\) level, and truncation of this series leads to our two asymptotic models, which we call the quadratic and cubic h-models. These models are well posed in spaces of analytic functions. We prove error bounds for the difference between solutions of the h-models and the water wave system. We also show that the Craig–Sulem models of water waves can be obtained from our asymptotic procedure. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water wave system. For three very different examples, we show that the agreement between the model equations and the water wave solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.
- Numerical Study of the Second-Order Correct Hamiltonian Model for
Unidirectional Water Waves- Abstract: Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure and, in certain circumstances, is globally well posed. A fully discrete, numerical algorithm based on the Fourier-spectral method is developed and its convergence tested. We then use this algorithm to generate solitary-wave solutions to the model. While such waves are known to exist, exact formulas for them are not available. The heart of the paper is a sequence of numerical experiments aimed at understanding the stability of individual solitary waves, their interaction, and whether or not the model exhibits resolution of general initial data into solitary waves. A comparison is made between the first-order correct KdV–BBM models and the associated second-order correct equations. A number of tentative conjectures pertaining to the models are put forward on the basis of these experiments.
PubDate: 2019-05-01
DOI: 10.1007/s42286-019-00003-y
- Abstract: Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure and, in certain circumstances, is globally well posed. A fully discrete, numerical algorithm based on the Fourier-spectral method is developed and its convergence tested. We then use this algorithm to generate solitary-wave solutions to the model. While such waves are known to exist, exact formulas for them are not available. The heart of the paper is a sequence of numerical experiments aimed at understanding the stability of individual solitary waves, their interaction, and whether or not the model exhibits resolution of general initial data into solitary waves. A comparison is made between the first-order correct KdV–BBM models and the associated second-order correct equations. A number of tentative conjectures pertaining to the models are put forward on the basis of these experiments.
- On the Lagrangian Period in Steep Periodic Waves
- Abstract: We show that, for periodic steep waves at finite water depth, the excess Lagrangian period, \(\Delta T_\mathrm{{L}}\) , made nondimensional by the Eulerian period, \(T_0\) , is equal to the mean Lagrangian drift velocity, \(u_\mathrm{{L}}\) , made nondimensional by the wave speed, c. This result is exact within second-order theory. Measurements in wave tank obtain that the two quantities are approximately equal. The vertical mean of \(\Delta T_\mathrm{{L}}\) and \(u_\mathrm{{L}}\) are zero in the experiments. The vertically averaged Lagrangian period thus equals the Eulerian period. The experimental and theoretical vertical derivative of \(\Delta T_\mathrm{{L}}\) are the same in the middle of the water column but strongly differ above the bottom and below the wave surface. These differences are due to the secondary streaming effect in the viscous boundary layers in the experiments, at the bottom and free surface. Fully nonlinear calculations complement the investigation.
PubDate: 2019-03-11
DOI: 10.1007/s42286-019-00004-x
- Abstract: We show that, for periodic steep waves at finite water depth, the excess Lagrangian period, \(\Delta T_\mathrm{{L}}\) , made nondimensional by the Eulerian period, \(T_0\) , is equal to the mean Lagrangian drift velocity, \(u_\mathrm{{L}}\) , made nondimensional by the wave speed, c. This result is exact within second-order theory. Measurements in wave tank obtain that the two quantities are approximately equal. The vertical mean of \(\Delta T_\mathrm{{L}}\) and \(u_\mathrm{{L}}\) are zero in the experiments. The vertically averaged Lagrangian period thus equals the Eulerian period. The experimental and theoretical vertical derivative of \(\Delta T_\mathrm{{L}}\) are the same in the middle of the water column but strongly differ above the bottom and below the wave surface. These differences are due to the secondary streaming effect in the viscous boundary layers in the experiments, at the bottom and free surface. Fully nonlinear calculations complement the investigation.