Three-dimensional Wave Patterns Research Articles

On nonlinear optical solitons of fractional Biswas-Arshed Model with beta derivative

The Biswas-Arshed (BA) model is an essential wave propagation model in nonlinear optics and optical fibers. In this manuscript, the enhanced G′/G-expansion method is taken into consideration. The BA model is used with this technique to locate visual soliton solutions. The solutions to the hyperbolic and trigonometric functions are determined using this technique. Each solution includes a large number of model and method parameters. The wave solutions are defined by the waves' character, which is determined by the three-dimensional (3D) and two-dimensional (2D) wave patterns with varying parameter values. The structure of the soliton shifts depending on the specific values of the parameters. These shifts are reflected in the 2D pictures that demonstrate the various soliton configurations. Finally, the obtained soliton solutions can be applied to high-speed data transmission to describe propagation pulses and to transmit light between two ends in optical fibers and communication systems.

Dynamics of closed-form invariant solutions and diversity of wave profiles of (2+1)-dimensional Ito integro-differential equation via Lie symmetry analysis

Nonlinear evolution equations are exploited in the models of complex nonlinear phenomena, explaining a number of our real-world problems in different fields of engineering and nonlinear sciences, for example, the modeling of interactions between ocean forces and the atmosphere, fiber optics, fluid dynamics, plasma physics, marine physics, and nonlinear dynamics. In this work, a comparatively new Lie symmetry analysis technique has been effectively implemented to explore some closed-form invariant solutions of the (2+1)-dimensional Ito equation. This technique is essential to minimize the number of independent variables by one at each stage until an ordinary differential equation (ODE) is formed. By utilizing the invariance analysis, the possible infinitesimal generators are derived, which are used to reduce the test problem and lead to determining the ODEs. All the exact solutions obtained are totally novel and differ significantly from previously published results. Due to the existence of some different arbitrary constants and functions, the solution provides a wealth of physical configurations. The solutions obtained in this article are thrust into the giant wave solutions such as curved shaped multi-solitons, parabolic waves, solitary waves, and traveling wave profiles displayed using three-dimensional (3D) wave patterns. Finally, we used symbolic computations in Mathematica to verify all the obtained soliton solutions. Furthermore, the current method’s superior expertise ensures that it is essentially capable of reducing the size of the computational assignment and can solve a variety of complex nonlinear partial differential equations (NLPDEs) that arise in many fields of nonlinear sciences, ocean engineering, nonlinear waves, and fluid dynamics.

Three-dimensional waves under ice computed with novel preconditioning methods

In this work, we present three-dimensional, nonlinear traveling wave solutions for water waves under a sheet of ice, i.e., flexural-gravity waves. The ice is modeled as a thin elastic plate on top of water of infinite depth and the equations are formulated as a boundary integral method. Depending on the velocity of the moving disturbance generating the flow, different deflection patterns of the floating ice sheet are observed. In order to compute solutions as efficiently as possible, we introduce a novel hybrid preconditioning technique used in an iterative Newton-Krylov solver. This technique is able to significantly increase the grid refinement and decrease the computational time of our solutions in comparison to methods that are presently used in the literature. We show how this approach is generalizable to three-dimensional ice wave patterns in different velocity regimes.

Synchronization-based reconstruction of electromechanical wave dynamics in elastic excitable media.

The heart is an elastic excitable medium, in which mechanical contraction is triggered by nonlinear waves of electrical excitation, which diffuse rapidly through the heart tissue and subsequently activate the cardiac muscle cells to contract. These highly dynamic excitation wave phenomena have yet to be fully observed within the depths of the heart muscle, as imaging technology is unable to penetrate the tissue and provide panoramic, three-dimensional visualizations necessary for adequate study. As a result, the electrophysiological mechanisms that are associated with the onset and progression of severe heart rhythm disorders such as atrial or ventricular fibrillation remain insufficiently understood. Here, we present a novel synchronization-based data assimilation approach with which it is possible to reconstruct excitation wave dynamics within the volume of elastic excitable media by observing spatiotemporal deformation patterns, which occur in response to excitation. The mechanical data are assimilated in a numerical replication of the measured elastic excitable system, and within this replication, the data drive the intrinsic excitable dynamics, which then coevolve and correspond to a reconstruction of the original dynamics. We provide a numerical proof-of-principle and demonstrate the performance of the approach by recovering even complicated three-dimensional scroll wave patterns, including vortex filaments of electrical excitation from within a deformable bulk tissue with fiber anisotropy. In the future, the reconstruction approach could be combined with high-speed imaging of the heart's mechanical contractions to estimate its electrophysiological activity for diagnostic purposes.

Metastable brain waves

Traveling patterns of neuronal activity—brain waves—have been observed across a breadth of neuronal recordings, states of awareness, and species, but their emergence in the human brain lacks a firm understanding. Here we analyze the complex nonlinear dynamics that emerge from modeling large-scale spontaneous neural activity on a whole-brain network derived from human tractography. We find a rich array of three-dimensional wave patterns, including traveling waves, spiral waves, sources, and sinks. These patterns are metastable, such that multiple spatiotemporal wave patterns are visited in sequence. Transitions between states correspond to reconfigurations of underlying phase flows, characterized by nonlinear instabilities. These metastable dynamics accord with empirical data from multiple imaging modalities, including electrical waves in cortical tissue, sequential spatiotemporal patterns in resting-state MEG data, and large-scale waves in human electrocorticography. By moving the study of functional networks from a spatially static to an inherently dynamic (wave-like) frame, our work unifies apparently diverse phenomena across functional neuroimaging modalities and makes specific predictions for further experimentation.

Numerical methods for modeling focused ultrasound in biomedical problems

The focused ultrasound (FUS) based on the cavitation effect is a promising technology that has a number of drawbacks, and the main one is the unpredictability of side effects. Numerical simulation of the influence of FUS on a human body can help to determine main mechanisms and factors that cause an undesirable tissue damage and adjust the medical equipment to avoid it. In this project, two numerical methods were used and compared within the scope of this problem. The grid-characteristic method allows to manage border and contact conditions of any complexity and implement a wide variety of damage models. The method was verified on experimental data from different areas of industry. In this project, it was used for model problems with simple geometry. The discontinuous Galerkin method is a stable and reliable numerical method that is less convenient in terms of border and contact conditions, but imposes fewer requirements on a computational grid. It allowed us to use grids based on real biological data. Both methods allow to obtain the full three-dimensional elastic wave pattern, including shear and surface waves, which are essential for understanding the influence of high-density inclusions on a FUS procedure. The applicability of both methods and their further development is discussed.

Wave Characteristics of Falling Film on Inclination Plate at Moderate Reynolds Number

Falling water film on an inclined plane is studied by shadowgraphy. The ranges of inclination angle and the film Reynolds number are, respectively, up to 21° and 60. Water is used as working fluid. The scenario of wave regime evolution is identified as three distinctive regimes, namely, initial quiescent smooth film flow, two-dimensional regular solitary wave pattern riding on film flow, and three-dimensional irregular wave pattern. Three characteristic parameters of two-dimensional solitary wave pattern, namely, inception length, primary pulse spacing, and propagation velocity, are examined, which are significant in engineering applications for estimation of heat and mass transfer on film flow. The present experimental data are well in agreement with the Koizumi correlations, the deviation from which is limited to 20% and 15%, respectively, for primary pulse spacing and propagation velocity. Through the scrutiny of the present experimental observation, it is concluded that wave evolution on film flow at the moderate Reynolds number is controlled by gravity and drag and the Rayleigh-Taylor instability that occurred on the steep front of primary pulse triggers the disintegration of continuous two-dimensional regular solitary wave pattern into three-dimensional irregular wave pattern.

Nonlinear dead water resistance at subcritical speed

The dead water resistance F1=12CdwρSU2 (ρ fluid density, U ship speed, S wetted body surface, Cdw resistance coefficient) on a ship moving at subcritical speed along the upper layer of a two-layer fluid is calculated by a strongly nonlinear method assuming potential flow in each layer. The ship dimensions correspond to those of the Polar ship Fram. The ship draught, b0, is varied in the range 0.25h0–0.9h0 (h0 the upper layer depth). The calculations show that Cdw/(b0/h0)2 depends on the Froude number only, in the range close to critical speed, Fr = U/c0 ∼ 0.875–1.125 (c0 the linear internal long wave speed), irrespective of the ship draught. The function Cdw/(b0/h0)2 attains a maximum at subcritical Froude number depending on the draught. Maximum Cdw/(b0/h0)2 becomes 0.15 for Fr = 0.76, b0/h0 = 0.9, and 0.16 for Fr = 0.74, b0/h0 = 1, where the latter extrapolated value of the dead water resistance coefficient is about 60 times higher than the frictional drag coefficient and relevant for the historical dead water observations. The nonlinear Cdw significantly exceeds linear theory (Fr < 0.85). The ship generated waves have a wave height comparable to the upper layer depth. Calculations of three-dimensional wave patterns at critical speed compare well to available laboratory experiments. Upstream solitary waves are generated in a wave tank of finite width, when the layer depths differ, causing an oscillation of the force. In a wide ocean, a very wide wave system develops at critical speed. The force approaches a constant value for increasing time.

Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton–Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

FORCES INDUCED ON A VERTICAL BREAKWATER BY INCIDENT OBLIQUE WAVES

Over the last years Navier-Stokes numerical models have been developed to accurately simulate wave interaction with all kinds of coastal structures, focusing on both functionality and stability of coastal structures. Although several models have been used to simulate wave interaction with coastal structures in two dimensions (2DV) there are a vast number of three-dimensional effects that need to be investigated in order to improve the design. In this paper a new model called IH-FOAM has been applied to study a vertical breakwater at prototype scale. As a first attempt of validation, the model has been used to simulate a regular wave train generated with a relative angle with the breakwater inducing three-dimensional wave patterns not only seaward the structure due to reflection but also generating an overtopping discharge variation along the breakwater trunk. Pressure laws and overtopping discharge at three different cross-sections along the structure have been studied. The pressure laws have been compared with classical Goda's formulation. Although, the numerical model predictions are in accordance with Goda's calculations, a clear three-dimensional variability of wave-induced pressure has been observed. Moreover, an additional study has been performed calculating pressure laws on the side-wall at the breakwater head. Large three-dimensional effects are detected from the simulations due to the flow separation at that area. Overtopping model predictions have been compared with Overtopping Manual calculations showing very close values along the trunk. However, lower overtopping discharge values are observed at the breakwater head. This paper is a preliminary work to show the range of applicability of a three-dimensional Navier-Stokes model to study wave interaction with a vertical breakwater under the action of an oblique wave train.

Patterned Substrates for the Study of Axonal Differentiation and Neuronal Response to Smooth Microtopography

The development of two and three-dimensional patterned substrates provides new ways to study neuronal behavior. The creation of two-dimensional (2D) patterned surfaces through the use of photolithography, microcontact printing, and self-assembled monolayer (SAM) chemistry allows for the study of axonal differentiation. Through the use of these methods, we have created starburst patterns to which E18 mouse hippocampal neurons are confined. Utilizing immunohistochemistry to specifically stain for the tau protein, predominantly localized along microtubules in the axon, we have found that neurite differentiation is not a predetermined process; rather, it is environmentally determined. The compilation of statistical data has shown that a mere 20μm difference in the distance a neurite is allowed to grow will determine polarization.Grayscale photolithography and solvent-assisted molding (SAMo) allow us to create smooth microtopography that mimics the microtopography encountered in vivo. We have fabricated continuous three-dimensional (3D) wave patterns, varying height as well as the peak-to-peak distance, to the study of neuronal behavior in response to smooth variations in microenvironment. This will provide insights into the limitations a neuron may experience during pathfinding in vivo.

Numerical Modeling of Tsunami Waves Interaction with Porous and Impermeable Vertical Barriers

Tsunami wave interaction with coastal regions is responsible for very important human and economic losses. In order to properly design coastal defenses against these natural catastrophes, new numerical models need to be developed that complement existing laboratory measurements and field data. The use of numerical models based on the Navier‐Stokes equations appears as a reasonable approach due to their ability to evaluate complex flow patterns around coastal structures without the inherent limitations of the classical depth‐averaged models. In the present study, a Navier‐Stokes‐based model, IH‐3VOF, is applied to study the interaction of tsunami waves with porous and impermeable structures. IH‐3VOF is able to simulate wave flow within the porous structures by means of the volume‐averaged Reynolds‐averaged Navier‐Stokes (VARANS) equations. The equations solved by the model and their numerical implementation are presented here. A numerical analysis of the interaction of a tsunami wave with both an impermeable and porous vertical breakwater is carried out. The wave‐induced three‐dimensional wave pattern is analysed from the simulations. The role paid by the porous media is also investigated. Finally, flow around the breakwater is analyzed identifying different flow behaviors in the vicinity of the breakwater and in the far field of the structure.

Frequency clusters and defect structures in nonlinear dust-density waves under microgravity conditions

Density waves in a dusty plasma emerge spontaneously at low gas pressures and high dust densities. These acousticlike wave modes were studied in a radio-frequency discharge under microgravity conditions. The complex three-dimensional wave pattern shows a spatially varying wavelength that leads to bifurcations, i.e., topological defects, where wave fronts split or merge. The calculation of instantaneous wave attributes from the spatiotemporal evolution of the dust density allows a precise analysis of those structures. Investigations of the spatial frequency distribution inside the wave field revealed that the wave frequency decreases from the bulk to the edge of the cloud in terms of frequency jumps. Between those jumps, regions of almost constant frequency appear. The formation of frequency clusters is strongly correlated with defects that occur exclusively at the cluster boundaries. It is shown that the nonlinearity of the waves has a significant influence on the topology of the wave pattern.

Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows

We study three-dimensional wave patterns on the surface of a film flowing down a uniformly heated wall. Our starting point is a model of four evolution equations for the film thickness h , the interfacial temperature theta , and the streamwise and spanwise flow rates, q and p , respectively, obtained by combining a gradient expansion with a weighted residual projection. This model is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary Marangoni effects for a wide range of parameters. For small Reynolds numbers, i.e., in the "drag-gravity regime," we observe regularly spaced rivulets aligned with the flow and preventing the development of hydrodynamic waves. The wavelength of the developed rivulet structures is found to closely match the one of the most amplified mode predicted by linear theory. For larger Reynolds numbers, i.e., in the "drag-inertia regime," the situation is similar to the isothermal case and no rivulets are observed. Between these two regimes we observe a complex behavior for the hydrodynamic and thermocapillary modes with the presence of rivulets channeling quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions, leading possibly to solitarylike waves. Two subregions are identified depending on the topology of the rivulet structures that can be either "ridgelike" or "groovelike." A regime map is further proposed that highlights the influence of the Reynolds and the Marangoni numbers on the rivulet structures. Interestingly, this map is found to be related to the variations of amplitude and speed of the two-dimensional solitary-wave solutions of the model. Finally, the heat transfer enhancement due to the increase of interfacial area in the presence of rivulet structures is shown to be significant.

Scroll Wave Filaments Terminate in the Back of Traveling Fronts

Experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction demonstrate that three-dimensional scroll waves can rotate around filaments that end in the wake of a traveling excitation pulse. The vortex structures nucleate during the collision of three nonrotating excitation pulses. The nucleation process and the wave-termination of filaments are direct consequences of the system's anomalous dispersion relation. Vortex filaments are found to expand with about twice the speed of their anchoring wave fronts. Filament expansion is accompanied by the build-up of phase differences in spiral rotation creating strongly twisted wave structures. Experiments employ optical tomography for the reconstruction of the three-dimensional wave patterns.

Three-dimensional wave patterns in long air cavities on a horizontal plane

Significant drag reduction of large displacement vessels can be achieved by applying multi-wave air cavities arranged on the hull bottom. Waves generated on the air–water interface of air cavities impose requirements on the dimensions of a hull recess that accommodates the air cavity. An approximate model for calculating wave patterns in the critical upstream part of long air cavities in a simplified, horizontal-plane geometry is presented in this paper. The influence of the recess planform boundaries and other factors on the wave patterns is studied parametrically. Some hydrodynamic aspects of multi-wave air cavities are discussed.

Wave patterns in film flows: modelling and three-dimensional waves

In a previous work, two-dimensional film flows were modelled using a weighted-residual approach that led to a four-equation model consistent at order , accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments.The stability of two-dimensional travelling waves to three-dimensional pertur-bations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al.(1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

The Patterns of Surface Capillary-gravity Short-crested Waves with Uniform Current Fields in Coastal Waters

A fully three-dimensional surface gravity-capillary short-crested wave system is studied as two progressive wave-trains of equal amplitude and frequency, which are collinear with uniform currents and doubly-periodic in the horizontal plane, are propagating at an angle to each other. The first- and second-order asymptotic analytical solutions of the short-crested wave system are obtained via a perturbation expansion in a small parameter associated with the wave steepness, therefore depicting a series of typical three-dimensional wave patterns involving currents, shallow and deep water, and surface capillary waves, and comparing them with each other.

On the Badulin, Kharif and Shrira model of resonant water waves

This paper is a reappraisal of the Hamiltonian model derived by Shrira, Badulin and Kharif (BKS) for three-dimensional nonlinear water waves. The model was introduced in [J. Fluid Mech. 318 (1996) 375] in an effort to describe the formation of traveling waves with crescent-shaped features that arise from the instability of the Stokes wave train at moderately large steepness. There have been observations of such traveling waves in wave tank experiments by Su et al. [J. Fluid Mech. 124 (1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. Some of the regimes described in these papers are of lightly breaking waves, which are asymmetric, with all crescents facing forward. Other regimes that they observe apparently give rise to traveling waves which have asymmetric crescent-shaped features facing both forwards and backwards. We show that the BKS model describes the Stokes wave train and its loss of stability at moderate amplitudes as a Hamiltonian saddle-node bifurcation, which corresponds to the formation of a stable three-dimensional wave pattern which exhibits asymmetric crescent-shaped elements. The model also produces a family of solutions homoclinic to the unstable Stokes wave train, which surrounds the orbit of crescent-shaped wave patterns and which provides a mechanism for transition. Other traveling wave solutions of the BKS model having nonzero transverse momentum are good candidates for the skew wave patterns possessing characteristic hexagonal shaped structures separated by quiescent stripes which are produced to the sides of the experiments in wave tanks. The BKS model has solutions which satisfy two of the three characteristics specified in [J. Fluid Mech. 318 (1996) 375] for nonlinear crescent-shaped waves, avoiding the introduction of a dissipative mechanism to describe features of these familiar wave patterns. The one weakness of the BKS model is that the crescent-shaped wave patterns are transformed to themselves under time reversal composed with a phase shift. Therefore all of the wave patterns described by the BKS model possess forward and backward facing crescent-shaped elements simultaneously, associated with alternating crests. These solutions reproduce the features of some but not all of the wave patterns in the observations of Su et al. [J. Fluid Mech. 124 (1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. In the deep water case, we introduce and analyze a new and more realistic four degrees of freedom Hamiltonian model of water waves which has two principal five wave interactions. While being more complicated and not completely integrable, nonetheless this model has traveling wave solutions with similar crescent-shaped elements, and others with the hexagonal features of the BKS model.

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.