Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

Phase velocity spectrum of internal waves in a weakly-stratified two-layer fluid

The problem of steady-state internal waves in a weakly stratified two-layer fluid with a density that is constant in the lower layer and depends exponentially on the depth in the upper layer is considered. The spectral properties of the equations of small perturbations of a homogeneous piecewise-constant flow are described. A nonlinear ordinary differential equation describing solitary waves and smooth bores on the layer interface is obtained using the Boussinesq expansion in a small parameter.     

三维海洋内孤立波数值水槽造波研究

海洋内孤立波因其分布广泛和携带巨大能量,对于潜艇安全航行影响很大.本文采用有限体积自适应半结构多重网格法求解Navier-Stokes方程,并用VOF方法追踪两层流体界面,应用双推板造波法进行内孤立波数值造波,建立了两层流体中的内孤立波数值水槽.数值模拟结果证实了该数值水槽数值造波的有效性和可靠性,为后续研究打下了基础...     

On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves

The evolution of a three-dimensional packet of weakly non-linear internal gravity waves propagating obliquely at an arbitrary angle to the vertical line is considered. Two coupled non-linear equations connecting variations of a packet amplitude and induced flows are derived. three-dimensionality of the packet having been found responsible for the non-linearity of the system. Explicit formulae for the induced flow vertical component and the mean density field variation caused by packet propagation have been obtained. The plane wave is shown to be unstable at any arbitrary slope of the wave vector. The non-linear equation describing the evolution of the two-dimensional packet is derived in the subsequent order of the asymptotic scheme.It has been found possible for the packet to collapse. The collapse of internal waves packets may be one of the possible mechanisms of “blini”-shaped regions of mixed waters formation in the ocean.     

Measurements of two- and three-dimensional waves in a channel,including the vicinity of cut-off frequencies

Measurements of water wave profiles were performed in a rectangular flume equipped with a modular wavemaker. This particular wavemaker could generate both two- and three-dimensional waves. A method is proposed to evaluate quantitatively the deviations of a spacial flow field from the two-dimensional one. Plane propagating waves, as well as pure sloshing waves with their crests parallel to the walls, were generated in the flume. In all cases the measured amplitudes were compared against linear theory predictions.     

Quasitransverse shock waves in elastic media with an internal structure

We consider the structure of small-amplitude quasitransverse shock waves in a weakly anisotropic elastic medium which possesses an internal structure generating the wave dispersion. The dispersion is modeled by introducing terms with higher derivatives into the equations of the theory of elasticity, and the dissipation is represented by viscous terms. In one of the two possible cases treated below, the requirement that the discontinuity structure exist leads to a set of admissible discontinuities of complex structure. A considerable part of the shock adiabat consists of a set of short portions and separate points, the number of which increases as the viscosity decreases. This complex set of admissible discontinuities is the general case where the dispersion in the shock-wave structure is sufficiently strong. Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 174–180, March–April, 1999.     

Generation of beams of three-dimensional periodic internal waves by sources of various types

The energy and force characteristics of periodic internal wave beams in a viscous exponentially stratified fluid are analyzed. The exact solutions of linearized problems of generation obtained by integral transformations describe not only three-dimensional internal waves but also the associated boundary layers of two types. The solutions not containing empirical parameters are brought to a form that allows a direct comparison with experimental data for generators of various types (friction, piston, and combined) of rectangular or elliptic shape. The stress tensor and force components acting on the generator are given in quadratures. In the limiting cases, the solutions are uniformly transformed to the corresponding expressions for the problems in a two-dimensional formulation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 12–23, May–June, 2006     

Large-amplitude internal waves in a two-layer liquid

The aim of the investigation was to extend the narrow band method [3] to forced wave motions and to verify the results experimentally. Experimental data were used in constructing the theoretical model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 132–135, March–April, 1991.     

Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston

The structure of the wave field given by an exact solution of the linearized problem of radiation of three-dimensional periodic internal waves in a continuously stratified viscous fluid is analyzed numerically. The waves are generated by a piston, i.e., a disk lying on a fixed horizontal plane and oscillating in the vertical direction. The flow fields and the wave displacements are compared with the data of shadow visualization and measurements of the wave amplitudes made using a contact sensor. The calculated and observed wave patterns are in satisfactory agreement and the displacement distributions coincide correct to a fitting coefficient 0.7 < K < 1.1 characterizing the role of the nonlinear effects and other factors neglected in this model.     

Analysis of plane and cylindrical nonlinear hyperelastic waves in materials with internal structure

A comparative analysis of two types of hyperelastic waves—plane waves (with plane front) and cylindrical waves (with curved front)—is offered. The propagation of the waves is studied theoretically for quadratically nonlinear hyperelastic media and numerically for a class of unidirectional fibrous composite materials. Hyperelasticity is described using the classical Murnaghan potential and a structural model of the first order—the model of effective constants. The internal structure of materials is described by this model and is at the micro-or nanolevels in numerical analysis. Particular attention is given to the evolution of the wave profile. It is studied in three stages: (i) derivation of nonlinear wave equations, (ii) construction of solutions in the form of plane and cylindrical waves, and (iii) numerical analysis of the evolution of these waves in composites with microlevel (Thornel) or nanolevel (Z-CNT) fibers. The main similarities and differences between plane longitudinal and cylindrical waves are shown. The most unexpected result is the striking difference between the evolution patterns numerically observed for plane and cylindrical wave profiles __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 21–46, October 2006.     

On a class of internal solitary waves in a two-layer fluid

Solitary waves on an interface between two fluids are considered. A uniform asymptotic expansion is constructed for internal solitary waves with flat crests (of the plateau type) that degenerate into a bore in the limit. It is shown that, in this case, in contrast to a Korteweg-de Vries wave, the wave amplitude is of the same order of smallness as the longwave approximation parameter. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 55–61, September–October, 1999.     

Harmonic waves in three-dimensional elastic composites

For a periodic elastic composite which consists of a matrix and fibers with finite dimensions (i.e. a three-dimensional problem), here are given estimates for eigenfrequencies and eigenfunctions. Calculations are based on a new quotient which has been proposed by Nemat-Nasser. The periodic character of the eigenfrequencies is pointed out, and illustrative examples are given.     

Parametric excitation of internal waves in a continuously stratified liquid

A study is made of the parametric excitation of internal waves in a continuously stratified liquid in a vessel executing oscillations in the vertical direction. It is shown that vertical oscillations of the vessel will excite oscillatory modes with eigenfrequencies equal to half of the oscillation frequency of the vessel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 167–169, September–October, 1982.I thank S. V. Nesterov for interest in the work.     

Generation of mode-2 internal waves in a two-dimensional stratification by a mode-1 internal wave

The generation of mode-2 nonlinear internal waves (IWs) by the evolution of a mode-1 IW in a two-dimensional stratification is investigated. A generation model accounting for intermodal interaction is derived based on a multi-modal approach in a weakly nonlinear and non-hydrostatic configuration. The generation model is numerically solved to simulate the evolution of mode-1 and mode-2 IWs in an inhomogeneous pycnocline. The numerical experiments confirm that mode-2 IWs are generated due to linear and nonlinear intermodal interaction. The mode-2 IW continues growing and gradually separates with the mode-1 IW during the generation process. The numerical results suggest that the pycnocline strength or thickness prominently affects the generation of mode-2 IWs, followed by pycnocline depth. A weakening or thinning pycnocline favors the generation of mode-2 IWs by evidently enhancing linear and nonlinear intermodal interaction, whereas a shoaling pycnocline favors a rapid growth rate mainly by enhancing linear intermodal interaction. The wave amplitude of an initial mode-1 IW strongly affects the generation of mode-2 IWs and increasing it can noticeably enlarge mode-2 IWs.     

Viscous attenuation of solitary internal waves in a two-layer fluid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 51–55, July–August, 1988.     

Visualization of turbulence and internal waves in a stratified fluid

A method for visualizing the velocity field in stratified flow by means of tracer particles made from wooden sawdust is described. The wood particles absorb water and salinity and may match the local density of stratified salt water.