Nonlinear Hydrodynamics Research Articles

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-wave is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are shown by the study of time variables. By virtue of the two breath-wave solutions, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

Control co-design optimization of nonlinear wave energy converters

This paper presents a study in which both the control and the shape of a Wave Energy Converter (WEC) are optimized simultaneously. A heaving point absorber WEC is assumed. To optimize the shape of the WEC’s buoy, nonlinear hydrodynamics need to be evaluated. One main contribution of this paper is the integration of nonlinear hydrodynamics and nonlinear control during the optimization of the WEC’s buoy shape. This approach is referred to as Control Co-Design (CCD). In this work, we present a control co-designed nonlinear heaving point absorber WEC that leverages the nonlinear dynamic, static Froude–Krylov (FK) forces to maximize power extraction. The nonlinear FK forces are approximated using a variation of the algebraic solution; the hydrodynamic forces of the body are computed using an analytic formulation leveraging the methods of eigenfunction expansion and separation of variables. The nonlinear geometry of the buoy is modeled as a series of inclined panels; the inclination angles are optimized to arrive at the optimal shape. The performance of the optimized shape is compared to that of a nonlinear spherical WEC. It is found that an average of 20% improvement is achieved by the optimized geometry over the spherical device.

A power-law model for nonlinear phonon hydrodynamics

A power-law model for nonlinear phonon hydrodynamics

Adaptive sampling physics-informed neural network method for high-order rogue waves and parameters discovery of the (2 + 1)-dimensional CHKP equation.

Rogue waves are important physical phenomena, which have wide applications in nonlinear optics, hydrodynamics, Bose-Einstein condensates, and oceanic and atmospheric dynamics. We find that when using the original PINNs to study rogue waves of high dimensional PDEs, the prediction performance will become very poor, especially for high-order rogue waves due to that the randomness of selection of sample points makes insufficient use of the physical information describing the local sharp regions of rogue waves. In this paper, we propose an adaptive sampling physics-informed neural network method (ASPINN), which renders the points in local sharp regions to be selected sufficiently by a new adaptive search algorithm to lead to a prefect prediction performance. To valid the performance of our method, the (2+1)-dimensional CHKP equation is taken as an illustrative example. Experimental results reveal that the original PINNs can hardly be able to predict dynamical behaviors of the high-order rogue waves for the CHKP equation, but the ASPINN method can not only predict dynamical behaviors of these high-order rogue waves, but also greatly improve the prediction efficiency and accuracy to four orders of magnitude. Then, the data-driven inverse problem for the CHKP equation with different levels of corrupted noise is studied to show that the ASPINN method has good robustness. Moreover, some main factors affecting the neural network performance are discussed in detail, including the size of training data, the number of layers of the neural network, and the number of neurons per layer.

Rings, Shells, and Arc Structures Around B[e] Supergiants. I. Classical Tools of Nonlinear Hydrodynamics

Structures in circumstellar matter reflect both fast processes and quasi-equilibrium states. A geometrical diversity of emitting circumstellar matter is observed around evolved massive stars, in particular around B[e] supergiants. We recapitulate classical analytical tools of linear and nonlinear potential theory, such as Cole–Hopf transformation and Grad–Shafranov theory, and develop them further to explain the occurrence of the circumstellar matter structures and their dynamics. We use potential theory to formulate the nonlinear hydrodynamical equations and test dilatations of the quasi-equilibrium initial conditions. We find that a wide range of flow patterns can basically be generated and the timescales can switch, based on initial conditions, and lead to eruptive processes, reinforcing that the nonlinear fluid environment includes both quasi-stationary structures and fast processes like finite-time singularities. Some constraints and imposed symmetries can lead to Keplerian orbits, while other constraints can deliver quasi-Keplerian ones. The threshold is given by a characteristic density at the stellar surface.

Breathers of Gerdjikov-Ivanov equation under background of elliptic function

As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB), and the general breather (GB). In recent years, the propagation of envelope breathers under the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers under the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been given under the elliptic function background. In this study, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions under the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, under the dn-periodic background we exhibit three types of interactions between two breathers, i.e. a GB and a KMB, an AB and a KMB, and a GB and an AB. Finally, we also present three types of interactions between two breathers under the general periodic background.

Nonlinear hydrodynamics of a heaving sphere in diffraction, radiation, and combined tests

We report on an experimental campaign designed to shed light on critical nonlinear hydrodynamic effects of heaving wave energy converter (WEC) buoys undergoing large-amplitude motions in operational conditions. The experiments carried out with a spherical model comprised radiation, diffraction and combined tests, where the vertical motion was prescribed and delivered via an actuator. As such, we had independent control of incident waves and motions, enabling isolation of different nonlinear terms by combining recordings from multiple phase- and amplitude-manipulated runs. All tests utilised short-duration wave groups and/or corresponding transient motion signals. We focus on analysis of nonlinear changes in the hydrodynamic forces, and free surface, in the first-harmonic frequency range - this is of most importance to WECs. In a series of radiation experiments, with progressively increasing imposed motions, the radiated wave field and the force in phase with the body velocity are found to decrease nonlinearly, pointing to the WEC's reduced ability to radiate waves under larger oscillations. In the combined tests, we are able to isolate various high-order cross-terms. We attempt to explain the observed trends through third-order potential flow interactions and consider a simple method to approximately describe these.

Numerical and Experimental Characterization of Rotational Floating Body Drag

Hydrodynamic drag plays a significant role in the motions and response of floating bodies – whether it be a wave energy converter, floating wind structure, or offshore oil & gas platform. Existing literature provides significant overview of the methodologies (both experimental and numerical) to characterize translational drag, however, there is limited research on the contributions (and methods of application) for rotational drag.
 
 This paper will detail both numerical modelling and a physical experimental campaign to assess how rotational drag impacts floating body dynamics, and best practices for numerical model inclusion. Specific focus will be on 1) the variety of methods used to input rotational drag into numerical models; 2) processes and lessons learnt from the experimental derivation of rotational drag coefficients; and 3) how does weakly non-linear wave stretching methods influence rotational drag.
 
 The experimental campaign is currently underway to classify the significance of rotational drag coefficients in characterizing floating body behavior. Translational and rotational drag coefficients of a simplified, inertial property matched, 1:50 floating body is being determined through a series of calibration tests. Both traditional free decay tests and forced oscillation tests will be implemented to evaluate these coefficients across multiple degrees-of-freedom. The final paper will present an overview of the experimental campaign, the results and lesson learnt.
 
 On the numerical side, the floating body will be modelled in the open-source wave energy converter modelling tool, WEC-Sim, and validated against the experimental results. Numerical results will be presented to review general body responses, with and without rotational drag, and generic wave conditions plus those expected at the PacWave wave energy test site in Oregon, USA.
 
 The inclusion of rotational drag coefficients and weakly nonlinear hydrodynamics are expected to improve computational model results, especially in the nonlinear wave excitation range, providing a better understanding of floating body behavior.

Hybrid linear potential flow - machine learning model for enhanced prediction of WEC performance

Numerical models based on the linear potential flow equations are of paramount importance in the design of wave energy converters (WECs). Over the years methods such as wave stretching, nonlinear Froude-Krylov and Morrison drag have been developed to overcome the short-comings of the underlying assumptions of small amplitude wave, small motion and inviscous flow. In this work we present a different approach to enhance the performance of the linear method: a hybrid linear potential flow – machine learning (LPF-ML) model. A hierarchy of high-fidelity models – Reynolds-Averaged Navier-Stokes, Euler and fully nonlinear potential flow – is used to create training data for correction factors targeting nonlinear hydrodynamics, pressure drag and skin friction, respectively. Long short-term memory (LSTM) networks are then trained and added to the LPF model. LSTM networks are heavy to train but fast to evaluate so the computational efficiency of the LPF model is kept high. Simple decay tests of generic bodies (sphere, box, etc) are used to validate the LPF-ML model. Finally, the LPF-ML is applied to a model-scale point-absorber WEC to assess the power production.

Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).

Model chromatin flows: numerical analysis of linear and nonlinear hydrodynamics inside a sphere.

We solve a hydrodynamic model of active chromatin dynamics, within a confined geometry simulating the cell nucleus. Using both analytical and numerical methods, we describe the behavior of the chromatin polymer driven by the activity of motors having polar symmetry, both in the linear response regime as well as in the long-term, fully nonlinear regime of the flows. The introduction of a boundary induces a particular geometry in the flows of chromatin, which we describe using vector spherical harmonics, a tool which greatly simplifies both our analytical and numerical approaches. We find that the long-term behavior of this model in confinement is dominated by steady, transverse flows of chromatin which circulate around the spherical domain. These circulating flows are found to be robust to perturbations, and their characteristic size is set by the size of the domain. This gives us further insight into active chromatin dynamics in the cell nucleus, and provides a foundation for development of further, more complex models of active chromatin dynamics.

Fully coupled aero-hydrodynamic modelling of floating offshore wind turbines in nonlinear waves using a direct time-domain approach

In standard operating conditions, a floating offshore wind turbine (FOWT) suffers from the combined action of nonlinear waves and wind. It is important to explore the effect of higher-order hydrodynamic loads induced by the nonlinear waves on its motion response. In this work, a fully coupled aero-hydrodynamic analysis method for FOWT is developed. The hydrodynamic part is based on the nonlinear potential flow theory and the perturbation approach, using a higher-order boundary element method in the time domain. Blade element momentum theory is applied to evaluate the aerodynamic load on the turbine rotor. The mooring system is modelled by the catenary theory. The developed method is verified against the published data of the NREL's 5-MW baseline wind turbine supported by the DeepCwind semi-submersible platform. The verifications are conducted stepwise in the following ways, including hydrodynamic wave excitation load, free-decay response, the load-displacement relationship of the mooring line, the steady-state response of the wind turbine, and fully coupled aero-hydrodynamic interaction. It is found that the nonlinear effect mainly influences the surge motion. The coupled configuration has little effect on the linear wave force and platform displacement but notably impacts the nonlinear wave force and displacement. The FOWT motion is found to be critical in changing the wave field, the maximum wave elevation position around the platform, and the nonlinear wave force acting on the floating platform. The results illustrate that the traditional indirect time-domain method using the Quadratic Transfer Function (QTF) based on assumed response cannot appropriately evaluate the nonlinear wave force of a FOWT in coupled motions. The results also reveal that nonlinear wave hydrodynamics is necessary to understand the motion response of a fully coupled FOWT.

Two-dimensional non-linear hydrodynamics and nanofluidics

A water monolayer squeezed between two solid planes experiences strong out-of-plane confinement effects while expanding freely within the plane. As a consequence, the transport of such two-dimensional water combines hydrodynamic and nanofluidic features, intimately linked with each other. In this paper, we propose and explicitly solve a non-linear hydrodynamic equation describing two-dimensional water flow with viscosity parameters deduced from molecular dynamic simulations. We demonstrate that the very ability of two-dimensional water to flow in short channels is governed by the second (dilatational) viscosity coefficient, leading to flow compression and velocity saturation in the high-pressure limit. The viscosity parameter values depend strongly on whether graphene or hexoganal boron nitride layers are used to confine 2D water that offers an interesting opportunity to obtain various nanofluids out of the same water molecules just by using alternate materials to fabricate the 2D channels.

Exact soliton solutions and stability analysis to (3 + 1)-dimensional nonlinear Schrödinger model

In this research, a dynamical (3 + 1)-dimensional nonlinear Schrödinger model (NLSM) is under consideration which is used to model the propagation of ultra-short optical pulses in highly-nonlinear media. This aforementioned model has been widely applied in Bose–Einstein condensates, nonlinear optical fiber communication plasma physics, hydrodynamics, elastic media and other diversified areas of mathematical physics and engineering. This study has two main objectives. Firstly, to obtain some novel solutions in the shape of singular, dark, periodic and plane wave solutions that are innovative and not previously found in the literature. Secondly, an amended extended tanh-function method with modulation instability (MI) is used for this model. As far as we know, this has never been investigated in this manner. The 2-D, 3-D and contour plots have been defined using the required parametric values to support the results of physical compatibility. According to the examined results, the method used in this study to retrieve consisting and conventional solutions is efficient and more immediate in computing and can be thought of as a useful tool in resolving more difficult problems that arise in the field of engineering, physics, mathematics and fiber optics.

Long-time correlations in a binary mixture: analysis of the nonlinearities of fluctuating-hydrodynamic equations

The equations of fluctuating nonlinear hydrodynamics (FNH) following the proper conservation laws are considered for a binary mixture. We focus on the density (ρ) correlations’ renormalization due to the FNH equations’ nonlinearities. The consequence of density nonlinearities treated in simplest approximations gives rise to the well-studied form of the mode coupling theory (MCT) for a two-component system. The MCT predicts a sharp ergodicity–nonergodicity transition similar to the one-component fluid in this idealized form. In the first part of the present paper, we compare the predictions of the idealized MCT model with the computer simulation results for a hard sphere mixture. We show that there is clear disagreement in long-time dynamic behaviour. Next, we consider the full set of nonlinearities in the FNH equations using a Martin–Siggia–Rose field theory. From the time reversal properties of the correlation and response function of the associated field theory, a set of fluctuation–dissipation relations (FDR) are obtained. These FDRs impose constraints on the long-time behaviour of the correlation functions. Our non-perturbative analysis considers the viability of freezing the time correlations for the two-component fluid over the longest time scale. Due to the FDR constraints arising from the nonlinearity in the FNH equations, a sharp ergodicity–nonergodicity transition for the binary mixture is not supported. If the are replaced as in terms of the average density ρ 0, ad hoc, while the density-nonlinearities in the pressure term of the corresponding FNH equations are kept, the ideal transition model of the simplified MCT is recovered.

Non-linear Eulerian hydrodynamics of dark energy: Riemann problem and finite volume schemes

Upcoming large-scale structure surveys can shed new light on the properties of dark energy. In particular, if dark energy is a dynamical component, it must have spatial perturbations. Their behaviour is regulated by the speed of sound parameter, which is currently unconstrained. In this work, we present the numerical methods that will allow to perform cosmological simulations of inhomogeneous dark energy scenarios where the speed of sound is small and non-vanishing. We treat the dark energy component as an effective fluid and build upon established numerical methods for hydrodynamics to construct a numerical solution of the effective continuity and Euler equations. In particular, we develop conservative finite volume schemes that rely on the solution of the Riemann problem, which we provide here in both exact and approximate forms for the case of a dark energy fluid.

Preface

International Workshop “Mathematical Modeling and Scientific Computing: Focus on complex processes and systems” (dedicated to the memory of Nikolai Botkin) Munich, October 4-7, 2022 This proceedings volume contains papers presented at the second International Workshop “Mathematical Modeling and Scientific Computing: Focus on complex processes and systems”, MMSC 2022, dedicated to the memory of Nikolai Botkin (22.03.1956–14.09.2019). Nikolai Botkin contributed significantly to mathematical modeling of various physical and biological processes, to the development of numerical methods and algorithms for optimal control of mechanical systems described by ordinary differential equations as well as for complex distributed systems, optimization and nonlinear programming, elasticity theory, hydrodynamics, thermodynamics, and homogenization theory.The first International Workshop MMSC 2020 was held on November 19-20, 2020 at the Technical University of Munich. The proceedings were published with CEUR-WS.org https://ceur-ws.org/Vol-2783/The second workshop MMSC 2022 was held on October 4-7, 2022 at the Technical University of Munich, Germany, in virtual format. Following the scientific interests of Nikolai Botkin, the workshop topics covered mathematical modeling and computer simulation for a wide range of optimal control and trajectory optimization problems, linear and nonlinear programming issues, as well as comprehensive applications in biology, medicine, thermodynamics and aerospace engineering. Recent achievements in machine learning and neural network technologies, challenges in simulation of cerebral blood circulation and bacterial quorum sensing were presented and discussed.In total, 35 invited talks of 77 authors from 12 countries were delivered. Out of 22 papers submitted to Morressier platform, 22 were accepted after peer-review.The editors would like to cordially thank the members of the Program and Organizing Committees, the participants, and the reviewers for actively contributing to the success of MMSC 2022.Editors of the MMSC 2022 ProceedingsVarvara Turova, Andrey Kovtanyuk, Johannes ZimmerList of Program Committee, Organizing Committee are available in this pdf.

Characterization and modeling of greenwater overtopping of a sea-level deck

Greenwater overtopping occurs when the deck of a marine structure or ship is submerged by a large non-breaking wave. This overtopping wave develops into a fast-moving mass of water that can damage deck structures. In this work we use laboratory experiments to characterize the kinematics of the greenwater fronts, showing that there are two main classes of flow. One class is an essentially two-dimensional flow with a single wave front traversing the entire deck, while the second type has multiple wave fronts originating from all sides of the platform. We characterize each type with respect to wave conditions and describe the transition between the two types. We then discuss two modeling approaches that can be used to simulate the kinematic of the overtopping fronts. A simple linear model, based on linear wave theory and on the assumption of the overtopping as a dam break, to predict the type and the propagation of the overtopping wave on the deck. The second is a three-dimensional nonlinear numerical hydrodynamics model, based on Smoothed Particle Hydrodynamics method, which reproduces the geometry of the overtopping in the laboratory more accurately. We compare the outputs of these models with laboratory observations and discuss their applicability.

Bifurcation, sensitivity analysis and exact traveling wave solutions for the stochastic fractional Hirota–Maccari system

The stochastic fraction Hirota–Maccari system is used as a governing model to check the wave propagation in nonlinear optics, plasma physics and hydrodynamics. Firstly, the random fractional order Hirota–Maccari system is transformed into a two-dimensional plane dynamic system by using traveling wave transformation. Secondly, the bifurcation is discussed by using the theory of plane dynamic system, and the sensitivity to the strength and frequency of perturbation is analyzed. In addition, the exact traveling wave solution of the stochastic fractional Hirota–Maccari system is constructed by using the analytical method of the plane dynamic system. Finally, the results of comparison analysis indicated that the fractional derivatives and noises will greatly influence the solution behavior.

Numerical prediction nonlinear heat-driven acoustics behaviours in standing-wave thermoacoustic engines using stress-blended Eddy simulation method

The present work investigated a standing-wave heat-driven thermoacoustic engine (SWTAE) system by capturing its nonlinear thermoacoustic features with three-dimensional (3D) unsteady Reynolds-Averaged Navier-Stokes (URANS) and hybrid URANS/LES (large Eddy simulation) models such as detached-Eddy simulations (DES) and stress-blended Eddy simulation (SBES) models. The comparison studies show that the prediction of the acoustic power of SWTAE using URANS is about 21.0% lower than that using LES, while the results from SBES and DES are relatively in good agreement with LES. Comparative studies of nonlinear hydrodynamics in the flow fields show that the results from SBES are closer to LES than DES, which can be attributed to the SBES model providing a faster transition to an explicit LES model outside the wall boundary layer. Furthermore, the heat transfer characteristics are compared by analyzing heat leaks and transversal heat flux, and it is found that the URANS model over-estimates the heat transfer characteristics, while the results of the other three models are relatively smaller than those obtained by URANS. In conclusion, the SBES model has great potential to be applied in simulating thermoacoustic nonlinear, heat transfer and flow behaviors of heat-driven SWTAEs.