Correlation between theoretical and experimental solitary waves

Experimental data on surface solitary waves generated by five methods are given. These data and literature information show that at amplitudes 0.2<a/h<0.6 (h is the initial depth of the liquid), experimental solitary waves are in good agreement with their theoretical analogs obtained using the complete model of liquid potential flow. Some discrepancy is observed in the range of small amplitudes. The reasons why free solitary waves of theoretically limiting amplitude have not been realized in experiments are discussed, and an example of a forced wave of nearly limiting amplitude is given. The previously established fact that during evolution from the state of rest, undular waves break when the propagation speed of their leading front reaches the limiting speed of propagation of a solitary wave is confirmed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 44–52, May–June, 1999.     

Velocity measurements for a turbulent nonseparated flow over solid waves

The laser-Doppler velocimeter was used to obtain measurements of the streamwise velocity over solid sinusoidal waves of small enough amplitude that a nonseparated flow existed. The measurements provide a critical test for Reynolds stress closure models since they are particularly sensitive to happenings in the viscous wall region (y ⁺ < 40), for which present theories are of uncertain accuracy. The results are compared with calculations that use an eddy viscosity model that successfully describes measurements of the wall shear stress along waves of small enough amplitude that a linear response is obtained. These calculations are in approximate agreement with measurements because they exactly account for inertia and viscous effects. However, there are significant differences which point to the inadequacy of turbulence models. In particular, non-linear effects and the amplitudes of the wave-induced velocity variations are underpredicted.     

Initial Conditions and Near-Field Dynamics in Turbulent Liquid Sheets

High-speed liquid “curtains” have been proposed to protect solid structures in fusion energy applications. Minimizing free-surface waves and spray formation in such flows is important for effective protection in this application. In this work, free-surface waves and turbulent breakup were studied experimentally in jets of water issuing from a rectangular nozzle into ambient air at a Reynolds number of 1.2 × 10⁵. Laser-Doppler anemometry was used to characterize the streamwise and transverse velocity components in the nozzle for two different flow calming section designs. Planar laser-induced fluorescence was used to measure the free-stream position in the near-field of the sheet. The results suggest that transverse velocity fluctuations in the nozzle are the primary factor in determining the amplitudes of free-surface waves. Removing a small amount of low-speed fluid immediately downstream of the nozzle exit (“boundary-layer cutting”) is shown to both significantly reduce free-surface waves and the amount of spray due to turbulent breakup. Overall, boundary-layer cutting appears to have the greatest benefit when used on a “well-conditioned” turbulent liquid sheet.     

A volume-agglomeration multirate time advancing for high Reynolds number flow simulation

A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Two examples of such situations are the travel of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, multistage explicit time advancing schemes with global time stepping are very accurate but very CPU consuming. In order to reduce this problem, the multirate (MR) time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions that would become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new MR scheme based on control volume agglomeration is proposed for the solution of the compressible Navier-Stokes equations equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macrocells for the smaller elements for stability purpose and a correction step in which small time steps are employed. The accuracy and efficiency of the proposed method are evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with a hybrid turbulence model of flows around bluff bodies like a flow around a space probe model at Reynolds number 10⁶, a circular cylinder at Reynolds number 8.4 × 10⁶, and two tandem cylinders at Reynolds number 1.66 × 10⁵ and 1.4 × 10⁵.     

Spectral method for solving the equal width equation based on Chebyshev polynomials

A spectral solution of the equal width (EW) equation based on the collocation method using Chebyshev polynomials as a basis for the approximate solution has been studied. Test problems, including the migration of a single solitary wave with different amplitudes are used to validate this algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The interaction of two solitary waves is seen to cause the creation of a source for solitary waves. Usually these are of small magnitude, but when the amplitudes of the two interacting waves are opposite, the source produces trains of solitary waves whose amplitudes are of the same order as those of the initial waves. The three invariants of the motion of the interaction of the three positive solitary waves are computed to determine the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Comparisons are made with the most recent results both for the error norms and the invariant values.     

The interactions between wave-currents and offshore structures with consideration of fluid viscosity

Study of the flow field around the large scale offshore structures under the action of waves and viscous currents is of primary importance for the scouring estimation and protection in the vicinity of the structures. But very little has been known in its mechanism when the viscous effects is taken into consideration. As a part of the efforts to tackle the problem, a numerical model is presented for the simulation of the flow field around a fixed vertical truncated circular cylinder subjected to waves and viscous currents based on the depth-averaged Reynolds equations and depth-averagedk-ɛ turbulence model. Finite difference method with a suitable iteration defect correct method and an artificial open boundary condition are adopted in the numerical process. Numerical results presented relate to the interactions of a pure incident viscous current with Reynolds numberRe=10⁵, a pure incident regular sinusoidal wave, and the coexisting of viscous current and wave with a circular cylinder, respectively. Flow fields associated with the hydrodynamic coefficients of the fixed cylinder, as well as corresponding free surface profiles and wave amplitudes, are discussed. The present method is found to be relatively straightforward, computationally effective and numerically stable for treating the problem of interactions among waves, viscous currents and bodies. The project supported by the National Natural Science Foundation of China and Foundation of State Key Laboratory of Ocean Engineering at Shanghai Jiao Tong University.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

影响大黄鱼的水中声指标

为了明确有效地评价声波对大黄鱼影响的指标,以水下爆破和滨海山体爆破两种作业方式产生的水中声波数据为基础,进行声波信号的声压峰值和声暴露级分析,并结合现场大黄鱼的行为响应进行研究。结果表明:水下爆破产生的水中声波瞬态声压幅值较高,能量在瞬间累积到最大值并影响大黄鱼的行为和安全;而滨海山体爆破水中声波的声压幅值小,声暴露时间长,能量的连续累积造成了大黄鱼行为的改变。由于两种爆破方式作用下的水中声波在不同的声暴露时间内的声暴露级较接近,且超过150 dB会造成大黄鱼的行为异常,因此认为声暴露级是评价水中声波对大黄鱼影响的一个重要指标。     

Propagation of small waves in inextensible strings

The model of an inextensible uniform string subject to constant gravitation is used to study the propagation of transversal waves in one-dimensional continua. Perturbation analysis of the equations of motion yields as a result the local representation of small waves in terms of a normalized Riemann function. By means of the latter, shape and speed of propagating waves may be discussed. A refined analysis confirms that on first order, small waves travel along characteristics of the unperturbed equilibrium configuration. An explicit power law for the waves’ amplitudes is given, and the findings are supported by the numerical results.     

On one-dimensional shock waves in composite materials modelled as interpenetrating solid continua

A one dimensional version of a theory of composite materials modelled as interpenetrating solid continua is used to study the propagation of shock waves in composites with two identifiable constituents. It is found that two distinct types of shock waves may propagate except when one of the constituents is a chopped fiber. The speeds at which the shock waves propagate are determined as are the differential equations which govern the evolutionary behaviour of the amplitudes of the waves. The implications of these results are studied in detail in a number of particular situations. Finally, the special results which hold when the amplitudes of the shock waves are infinitesimal are also presented.     

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

The compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number

This paper focuses on the assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. The Taylor–Green vortex at Re = 1600 is considered. The results are compared with those obtained using a pseudo‐spectral solver, converged on a 512³ grid and taken as the reference. The temporal evolution of the dissipation rate, visualisations of the vortical structures and the kinetic energy spectrum at the instant of maximal dissipation are compared to assess the results. At an effective resolution of 288³, the fourth‐order accurate discontinuous Galerkin method (DGM) solution (p = 3) is already very close to the pseudo‐spectral reference; the error on the dissipation rate is then essentially less than a percent, and the vorticity contours at times around the dissipation peak overlap everywhere. At a resolution of 384³, the solutions are indistinguishable. Then, an order convergence study is performed on the slightly under‐resolved grid (resolution of 192³). From the fourth order, the decrease of the error is no longer significant when going to a higher order. The fourth‐order DGM is also compared with an energy conserving fourth‐order finite difference method (FD4). The results show that, for the same number of DOF and the same order of accuracy, the errors of the DGM computation are significantly smaller. In particular, it takes 768³ DOF to converge the FD4 solution. Finally, the method is also successfully applied on unstructured high quality meshes. It is found that the dissipation rate captured is not significantly impacted by the element type. However, the element type impacts the energy spectrum in the large wavenumber range and thus the small vortical structures. In particular, at the same resolution, the results obtained using a tetrahedral mesh are much noisier than those obtained using a hexahedral mesh. Those obtained using a prismatic mesh are already much better, yet still slightly noisier. Copyright © 2013 John Wiley & Sons, Ltd.     

Phase-locking in a jet forced with two frequencies

This study investigates the effect of forcing a shear layer at more than one frequency. Multiple frequency forcing permits the phase and initial relative amplitudes among unstable waves to be manipulated. More control can be imposed on vortex merging and mixing. Various vortex merging modes were observed and explained by the relative strength of the instability waves and their phase alignment. The vortex phase and path jitterings present in single-frequency forcing cases are greatly reduced when forced at more than one frequency. The observed cycle-to-cycle variation was small. This enables phase-lock measurements to be performed more easily. The phase-lock data show excellent agreement with the flow visualization results even when averaged over only a few cycles.     

Nonlinear interaction of longitudinal—Transverse acoustic waves in a circular cylinder

The limiting amplitudes of acoustic oscillations in a cylindrical volume of a heat releasing medium in which one or several modes are unstable in the linear approximation are determined. One of the mechanisms limiting the amplitudes of unstable acoustic modes is the transfer of energy from them to damped modes by nonlinear interaction. The nonlinear interactions of plane acoustic waves in a long channel have been considered by Artamonov and Vorob'ev [1]; in the present paper, the interaction of mixed longitudinal—transverse acoustic modes in a closed cylindrical volume is considered. The equations describing the interaction of two and three longitudinal—transverse modes are derived and investigated in the quadratic approximation by the method of slowly varying amplitudes and phases of the oscillations [2]. The treatment is applicable to a high-temperature gas, for which general stability conditions in the linear approximation have been formulated by Artamonov [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–9, September–October, 1982.I should like to express my thanks to K. I. Artamonov (deceased) for suggesting the problem and for scientific supervision and A. P. Vorob'ev for constant interest in the work and helpful advice.     

Observations on waveforms of capillary and gravity-capillary waves

Due to extreme conditions in the field, there has not been any observational report on three-dimensional waveforms of short ocean surface waves. Three-dimensional waveforms of short wind waves can be found from integrating surface gradient image data (Zhang 1996a). Ocean surface gradient images are captured by an optical surface gradient detector mounted on a raft operating in the water offshore California (Cox and Zhang 1997). Waveforms and spatial structures of short wind waves are compared with early laboratory wind wave data (Zhang 1994, 1995). Although the large-scale wind and wave conditions are quite different, the waveforms are resoundingly similar at the small scale. It is very common, among steep short wind waves, that waves in the capillary range feature sharp troughs and flat crests. The observations show that most short waves are far less steep than the limiting waveform under weak wind conditions. Waveforms that resemble capillary-gravity solitons are observed with a close match to the form theoretically predicted for potential flows (Longuet-Higgins 1989, Vanden-Broeck and Dias 1992). Capillaries are mainly found as parasitic capillaries on the forward face of short gravity waves. The maximum wavelength in a parasitic wave train is less than a centimeter. The profiles of parasitic wave trains and longitudinal variations are shown. The phenomenon of capillary blockage (Phillips 1981) on dispersive freely traveling short waves is observed in the tank but not at sea. The short waves seen at sea propagate in all directions while waves in the tank are much more unidirectional.     

Propagation of weakly nonlinear disturbances of the interface between two layers of fluid

The dynamics of internal waves of small but finite amplitude in a two-layer fluid system bounded by rigid horizontal surfaces at bottom and top is investigated theoretically. For linear disturbances of the fluid interface the authors propose a polynomial approximation of the dispersion relation which has the same asymptotics as the exact formula in the limiting situations of very long and short waves. In the case of three-dimensional, weakly nonlinear disturbances of slowly varying shape (in the coordinate system moving with the wave) an equation like the wave equation is derived. This equation has Stokes solutions coinciding with the well-known results for infinitely deep layers. For fairly long disturbances solitary solutions of the model wave equation which fit the experimental data are determined. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 125–131, January–February, 1994.     

On solitary waves in plasma

From the equations of hydrodynamics and electrodynamics, a system of coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exists for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The interaction of the longitudinal solitary waves with each other is studied and an upper bound on the amplitudes of these waves is obtained. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.     

A method for the calculation of nonsymmetric steady periodic capillary–gravity waves on water of arbitrary uniform depthUne méthode de calcul des ondes de gravité–capillarité périodiques et non-symétriques en profondeur arbitraire

In this Note the method developed by Aider and Debiane (2004) for the calculation of nonsymmetric water waves on infinite depth is extended to finite depth. The water-wave problem is reduced to a system of nonlinear algebraic equations which is solved by using Newton's method. Solutions are computed up to their limiting forms by decrementing the depth from the infinity to a value of the depth-wavelength ratio $h/\lambda$ less than 0.025. It is found that the waves become symmetric when the depth becomes very small. Relations giving some integral properties are derived. To cite this article: R. Aider, M. Debiane, C. R. Mecanique 334 (2006).     

Wave propagation in heterogeneous anisotropic plates involving large deflection

Equations of motion for laminated plates of composite materials are derived for motions of large amplitudes as well as for incremental deformations superposed on large deflections. Free waves of large and small amplitudes propagating, respectively, in initially flat and deformed plates are investigated. By using a special substitution for the extensional displacement components the governing equations appear to be linear and consequently the analysis of free wave propagation greatly simplified. It is found that a train of free wave consists of a finite number of simple harmonic waves.     

Comments on the relation between surface wave theory and the theory of reflection

The sextic Stroh formalism, previously extensively used in the analysis of subsonic phenomena, has been used for the analysis of reflection phenomena and leaky surface waves in the first transsonic range of velocities. In particular the behaviour of the reflection problem at the limiting velocity is studied. It is shown that when the condition of free surface can be satisfied without the inhomogeneous partial wave, a situation which would appear to be the natural limiting case of a surface wave of infinite penetration, the body wave alone satisfies the condition of free surface. This result illuminates the Barnett-Lothe existence theorem for subsonic surface waves. The close connection between the reflection problem and the leaky surface wave problem becomes very apparent in the formalism used. It is shown that for a point on a branch of leaky waves where the solution is undamped, the conditions for simple reflection, i.e. reflection only involving the two body waves, are also present. In the vicinity of such a point reflection is accompanied by resonance excitation of leaky waves. The paper concludes with some explicit calculations for transversely isotropic solids.