Search Results (23)

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, $Q\left(t,x\right)=A_g{u}^m,m\ge 1$. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for $m=3$, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form. Full article

Oil-spill booms in shallow waters and high tidal amplitudes could ground on the seabed and retain high amounts of seawater. The object of this study is to estimate the mooring force at both boom section ends and the occurrence of submarining observed along the crest line. We use a Lagrangian linear elastic membrane theory incorporating the non-linear Green strain tensor and a non-updated hydrostatic or hydrodynamic load. We describe a numerical method using geometrically non-linear finite elements and 2D vertical hydrostatic pressure estimation. The calculated results indicate the role of hydrostatic pressure caused by the water height difference—several centimeters at the mid-section—and the influence of the elasticity module. We consolidate the mooring force results by supposing 2D horizontal hydrodynamic pressure. We associate the current velocity that produces the same mooring force with that generated by the hydrostatic load. The associated Froude number is close to 0.8. Full article

This study examines the evolution characteristics of ship waves generated by large vessels in shallow waters. A CFD-based numerical wave tank, incorporating Torsvik’s ship wave theory, was developed using the VOF multiphase approach and the RNG k-ε turbulence model to capture free-surface evolution and turbulence effects. Results indicate that wave heights vary significantly near the critical depth-based Froude number ($F_h$). Comparative analyses between CFD results for a Wigley hull and proposed empirical correction formulas show strong agreement in predicting maximum wave heights in transcritical and supercritical regimes, accurately capturing the nonlinear surge of wave amplitude in the transcritical range. Simulations of 2000-ton and 6000-ton class vessels further reveal that wave heights increase with $F_h$, peak in the transcritical regime, and subsequently decay. Lateral wave attenuation was also observed with increasing transverse distance, highlighting the role of vessel dimensions and bulbous bow structures in modulating wave propagation. These findings provide theoretical and practical references for risk assessment and navigational safety in shallow waterways. Full article

We study here the Boussinesq-type Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation, for describing the wave propagation of long, nonlinear, and dispersive waves in shallow water and other physical scenarios. In order to obtain novel families of wave solutions, we apply two efficient analytical techniques: the Modified Extended tanh (ME-tanh) method and the Modified Residual Power Series Method (mRPSM). These methods are used for the very first time in this equation to produce both exact and high-order approximate solutions with rich wave behaviors including soliton formation and energy localization. The ME-tanh method produces a rich class of closed-form soliton solutions via systematic simplification of the PDE into simple ordinary differential forms that are readily solved, while the mRPSM produces fast-convergent approximate solutions via a power series representation by iteration. The accuracy and validity of the results are validated using symbolic computation programs such as Maple and Mathematica. The study not only enriches the current solution set of the Boussinesq-type KdV equation but also demonstrates the efficiency of hybrid analytical techniques in uncovering sophisticated wave patterns in multimensional spaces. Our findings find application in coastal hydrodynamics, nonlinear optics, geophysics, and the theory of elasticity, where accurate modeling of wave evolution is significant. Full article

This work aims to highlight the effects of nonlinearity on the crest shape of large directional water wave events. To simulate such events, we chose to focus frequencies on a pre-determined time step over a wavefield with randomised phases, running the simulations with HOS-ocean, a fully nonlinear potential flow solver. By also applying a phase separation scheme, we were able to identify the contributions of the various orders of nonlinearity to the formation of these large wave events. The findings show a significant change in the shape of these large water waves compared to linear theory, particularly in shallower water depth. In addition, the phase separation reveals the increased significance of high-order harmonics in finite water depths compared to deep water. Full article

This study presents a one-dimensional solver of the shallow water equations designed for the wet-bed step Riemann problem. Nonlinear mass and momentum equations incorporating shock and rarefaction waves in a straight one-dimensional channel are expressed as a pair of equations that depend solely on local depth values either side of the step. These unified equations are uniquely designed for the four conditions involving shock and rarefaction waves that can occur in the Step Riemann Problem. The Levenberg–Marquardt method is used to solve these simplified nonlinear equations. Four verification tests are considered for shallow free surface flow in a wet-bed channel with a step. These cases involve two rarefactions, opposing shock-like hydraulic bores, and a rarefaction and shock-like bore. The numerical predictions are in close agreement with existing theory, demonstrating that the method is very effective at solving the wet-bed step Riemann problem. Full article

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs. Full article

The problem of creating methods for calculating tsunami parameters and predicting these dangerous events is currently being solved by integrating the equations of the theory of water waves. Both numerical methods and powerful computers are used, as well as analytical solutions. The essential stage is the stage of the tsunami reaching the shelf and shallow coastal waters. The tsunami amplitude increases here, and nonlinear effects become important. Nonlinearity excludes the solution’s unicity and the superposition principle’s fulfillment. The nonlinear theory can have many solutions, depending on various external conditions; there could be nontrivial ones. In this article, we explore the properties of several nonlinear solutions. With their help, we can find the maximum possible amplitude of tsunami waves when approaching the coast and estimate the seismological parameters of an earthquake-generating tsunami. Full article

This study is concerned with the generation and propagation of strongly nonlinear waves in shallow water. A numerical wave flume is developed where nonlinear waves of solitary and cnoidal types are generated by use of the Level I Green-Naghdi (GN) equations by a piston-type wavemaker. Waves generated by the GN theory enter the domain where the fluid motion is governed by the Navier–Stokes equations to achieve the highest accuracy for wave propagation. The computations are performed in two dimensions, and by an open source computational fluid dynamics package, namely OpenFoam. Comparisons are made between the characteristics of the waves generated in this wave tank and by use of the GN equations and the waves generated by Boussinesq equations, Laitone’s 1st and 2nd order equations, and KdV equations. We also consider a numerical wave tank where waves generated by the GN equations enter a domain in which the fluid motion is governed by the GN equations. Discussion is provided on the limitations and applicability of the GN equations in generating accurate, nonlinear, shallow-water waves. The results, including surface elevation, velocity field, and wave celerity, are compared with laboratory experiments and other theories. It is found that the nonlinear waves generated by the GN equations are highly stable and in close agreement with laboratory measurements. Full article

The purpose of this paper is to propose the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. We begin with the one-dimensional nonlinear shallow-water wave equations, which we consider over a beach of complex geometry that can be modeled by a piecewise continuous function, along with several natural initial and boundary conditions. The primary obstacle in solving this problem is the moving boundary associated with the shoreline motion. To avoid this difficulty, we replace the moving boundary with a stationary boundary by applying a transformation to the spatial variable of the computational domain. A characteristic feature of any tsunami problem is the smallness of the parameter $\epsilon ={\eta }_0/h_0$, where ${\eta }_0$ is the characteristic amplitude of the wave, and $h_0$ is the characteristic depth of the ocean. The presence of this small parameter enables us to effectively linearize the problem by using the method of perturbations, which leads to an analytical solution via an integral transformation. This analytical solution assumes that there is no wave breaking. In light of this assumption, we introduce the wave no-breaking criterion and determine bounds for the applicability of our theory. The proposed model can be readily used to investigate the tsunami run-up and draw-down for different sea bottom profiles. The novel particular solution, when the seafloor is described by the piecewise linear function, is obtained, and the effects of the different beach profiles and initial wave locations are considered. Full article

The understanding of the occurrence of extreme waves is crucial to simulate the growth of waves in coastal regions. Laboratory experiments were performed to study the spatial evolution of the statistics of group-focused waves that have a relatively broad-banded spectra propagating from intermediate water depth to shallow regions. Breaking waves with different spectral types, i.e., spectral bandwidths and wave nonlinearities, were generated in a wave flume using the dispersive focusing technique. The non-Gaussian behavior of the considered wave trains was demonstrated by the means of the skewness and kurtosis parameters estimated from a time series and was compared with the second-order theory. The skewness and kurtosis parameters were found to have an increasing trend during the focusing process. During both the downstream wave breaking and defocusing process, the wave train dispersed again and became less steep. As a result, both skewness and kurtosis almost returned to their initial values. This behavior is clearer for narrower wave train spectra. Additionally, the learning algorithm multilayer perceptron (MLP) was used to predict the spatial evolution of kurtosis. The predicted results are in satisfactory agreement with experimental findings. Full article

An overview is presented of several diverse branches of work in the area of effectively 2D fluid equilibria which have in common that they are constrained by an infinite number of conservation laws. Broad concepts, and the enormous variety of physical phenomena that can be explored, are highlighted. These span, roughly in order of increasing complexity, Euler flow, nonlinear Rossby waves, 3D axisymmetric flow, shallow water dynamics, and 2D magnetohydrodynamics. The classical field theories describing these systems bear some resemblance to perhaps more familiar fluctuating membrane and continuous spin models, but the fluid physics drives these models into unconventional regimes exhibiting large scale jet and eddy structures. From a dynamical point of view these structures are the end result of various conserved variable forward and inverse cascades. The resulting balance between large scale structure and small scale fluctuations is controlled by the competition between energy and entropy in the system free energy, in turn highly tunable through setting the values of the conserved integrals. Although the statistical mechanical description of such systems is fully self-consistent, with remarkable mathematical structure and diversity of solutions, great care must be taken because the underlying assumptions, especially ergodicity, can be violated or at minimum lead to exceedingly long equilibration times. Generalization of the theory to include weak driving and dissipation (e.g., non-equilibrium statistical mechanics and associated linear response formalism) could provide additional insights, but has yet to be properly explored. Full article

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy. Full article

Numerical models are useful tools for studying complex wave–wave and wave–current interactions in coastal areas. They are also very useful for assessing the potential risks of flooding, hydrodynamic actions on coastal protection structures, bathymetric changes along the coast, and scour phenomena on structures’ foundations. In the coastal zone, there are shallow-water conditions where several nonlinear processes occur. These processes change the flow patterns and interact with the moving bottom. Only fully nonlinear models with the addition of dispersive terms have the potential to reproduce all phenomena with sufficient accuracy. The Boussinesq and Serre models have such characteristics. However, both standard versions of these models are weakly dispersive, being restricted to shallow-water conditions. The need to extend them to deeper waters has given rise to several works that, essentially, add more or fewer terms of dispersive origin. This approach is followed here, giving rise to a set of extended Serre equations up to kh ≈ π. Based on the wavemaker theory, it is also shown that for kh > π/10, the input boundary condition obtained for shallow-waters within the Airy wave theory for 2D waves is not valid. A better estimate for the input wave that satisfies a desired value of kh can be obtained considering a geometrical modification of the conventional shape of the classic piston wavemaker by a limited depth $\theta h$, with $\theta \le$ 1.0. Full article

The Korteweg–de Vries equation (KdV) is a mathematical model of waves on shallow water surfaces. It is given as third-order nonlinear partial differential equation and plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, and a detailed analysis was performed by Korteweg and de Vries in 1895. In this article, by using multi-linear estimates in Bourgain type spaces, we prove the local well-posedness of the initial value problem associated with the Korteweg–de Vries equations. The solution is established online for analytic initial data $w_0$ that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improves upon earlier results. Full article