The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

An analytical solution of a linearized problem of the emission of periodic internal waves by part of a plane which oscillates with a small amplitude in an arbitrary direction in a viscous exponentially stratified fluid is constructed. Solutions of the dispersion equation are given for all positions of the emitting surface (arbitrary, vertical, horizontal, and critical when one of the beam propagation directions is collinear with the emitting surface). The possibility of transition to the case of a uniform fluid, which is important for applications, is analyzed.     

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     

Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid

The problem of the stabilization of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid is solved using both asymptotic and numerical methods. The analytical solution describes the structure of the main convective cells, including thin meridional jets flowing along the surface and plumes spreading from the flow convergence regions above the upper and lower poles of the sphere which gradually return the fluid particles to the neutral buoyancy horizon. The total width of the flows adjacent to the surface exceeds the thickness of the salinity deficit layer or the density boundary layer. The numerical solution of the complete problem in the nonlinear formulation describes the main convective cells and two systems of unsteady integral waves formed in the vicinity of the sphere poles. At large times, out of the entire system of internal waves only those nearest to the neighborhood of their horizon of formation remain clearly defined. The calculated flow patterns are in agreement with each other and the data of shadow visualization of the stratified fluid structure near a submerged obstacle at rest.     

Generation of Periodic Motions by a Disk Performing Torsional Oscillations in a Viscous, Continuously Stratified Fluid

The solution of the problem of nonlinear generation of periodic internal waves by a boundary flow on a vertical cylinder or a horizontal disk performing torsional oscillations in an exponentially stratified fluid is constructed. The calculations are in satisfactory agreement with the results of experiments in which both horizontal and inclined disks of various diameters and a model propeller performing periodic torsional oscillations, including oscillations against a background of uniform rotation, are used as perturbation sources. The experiments were carried out over a wide range of parameters including the laminar, transition, and turbulent flow regimes. The limits of applicability of the proposed analytic theory of wave radiation are determined.     

Numerical modeling of the generation of internal waves by uniform stratified flow over a thin vertical barrier

Numerical simulations of two‐dimensional stratified flow past an obstacle (thin vertical strip) were performed at relatively low Reynolds numbers. A finite differences solver was adopted to simultaneously solve Navier–Stokes equations together with transport equations for salinity (stratifying agent), and the standard Smagorinsky turbulent closure scheme was called in whenever necessary to account for turbulence. The emphases were on the evaluation of code for unsteady stratified flow applications as well as identification of transient and steady internal‐wave processes during flow past obstacles. Simulations were compared with laboratory experiments, where observations were made using a high resolution Schlieren technique and conductivity probes. Blocking was observed upstream of the obstacle, surrounded by near‐zero frequency internal waves, the phase lines of which joined those of lee waves through a transition zone in the proximity of the obstacle. This pattern was preceded by initial transients of the starting flow in which propagating internal waves played a dominant role. Confluence of isopycnals passing over/under the obstacle in the wake led to interesting flow phenomena, including the radiation of internal waves. The numerical simulations were in good agreement with observations, except that some phenomena could not be captured due to resolution issues of either numerical or experimental techniques. The efficacy of the code in point for stratified flow calculations with relevance to the atmosphere and oceans was confirmed. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

One-dimensional nonlinear induced dynamics of acoustic waves in a finite three-dimensional domain

The problem of the time-dependent viscous compressible gas flow excited by a small external time-dependent space-and time-periodic force is considered within the framework of the Navier-Stokes equations on a finite interval with periodic boundary conditions. The investigation is carried out numerically for a periodicity interval L, divided by the viscous length, from 10² to 2 × 10³ and external force amplitudes from 10^?4 to 0.1. The nonlinear dynamics of the wave processes are investigated within the framework of this problem. It is shown that nonlinear steady-state oscillations with sharp variation of the quantities in space and time develop when L is greater than or of the order of 10³. This leads to the onset of a continuous spectrum.     

Transformation of hanging discontinuities into vortex systems in a stratified flow behind a cylinder

The mechanisms of formation of hanging discontinuities, vortex dipoles, and vortex arrays in the wave wake behind a cylinder moving at a constant velocity in a stratified fluid are investigated using various schlieren methods. The existence of discontinuities is attributable to the distortion of the internal-wave phase pattern in the shear flow and to the varying stratification and subsequent interaction of the waves with the appearing nonuniformities. Hanging discontinuities and vortex systems are low-velocity analogs of shock waves. An analysis of the internal-wave pattern indicates that the values of the normal velocity component differ on the upper and lower edges of the discontinuities. A regime diagram for flows of this kind is given.     

Experiments on the interaction of internal waves with the semi-submersible platform in a stratified fluid

对密度分层流体中内波与半潜平台的相互作用问题 进行了模型试验研究. 采用摇板方法进行了内波造波试验,对内波波长、周期和波高进行了 测量分析,获得了内波波高和波长与周期之间的相关关系. 利用激光和倾角仪方法对半潜平 台的纵荡和纵摇运动响应进行了测量分析,获得了平台纵荡运动及纵摇角幅值与内波周期之 间的相关关系. 结果表明,在半潜平台的设计与应用中,内波对其运动响应的影响是不可忽 视的. 特别地,发现了在内波周期的某个范围内, 半潜平台的纵荡和纵摇都会出现 倍频周期运动响应的现象.     

Generation of internal waves by a turbulent jet in a stratified fluid

The process of generation of internal waves by an initially cylindrical, turbulent jet with a Gaussian profile of the average horizontal velocity component in a fluid with stable linear density stratification is investigated by direct numerical simulation. It is shown that on time intervals Nt < 30, where N is the buoyancy frequency, the vertical velocity pulsations collapse, which is accompanied by the generation of internal waves whose spatial period is close to the wavelength of the spiral mode of jet instability in a homogeneous fluid. The wave dynamics and kinematics can be satisfactorily described by the linear theory for a pulsed source and their parameters are in good agreement with the parameters of the “coherent” internal waves generated by a stratified wake in a laboratory experiment. At large times the wave generation ceases and the variations of the fluid density are localized in the neighborhood of the centers of large-scale vortices formed in the horizontal plane in the neighborhood of the jet.     

Symmetry of a periodic array of particles and a viscous fluid flow in the stokes approximation

A suspension in which rigid spherical particles of the same radius form a periodic array is considered. A general solution of the Stokes equations periodic with respect to this array is obtained. With reference to a fluid flow through a fixed array and a shear flow with frozen-in particles it is shown that taking the array structure and the symmetry of the conditions on the particle surface into account leads to a considerable simplification of the problem and makes it possible to determine the velocity and pressure distributions over the fluid.     

Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity

We prove the existence of multidimensional traveling wave solutions of the bistable reaction-diffusion equation with periodic coefficients under the condition that these coefficients are close to constants. In the case of one space dimension, we prove their asymptotic stability.     

Conjugate shear flows of a weakly stratified fluid

This paper studies the problem of pairs of horizontal shear flows of weakly stratified fluids with identical mass, momentum, and energy fluxes. The initial problem is reduced to a system of two scalar equations for the main- and perturbed-flow parameters by using bifurcation methods. The existence conditions for nontrivial branches of conjugate flows close to the main flow are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 79–88, March–April, 2009.     

Breaking and cascade of internal gravity waves in a continuously stratified fluid

The evolution of a few large scale high frequency standing internal waves confined to a vertical plane is studied numerically. The growth of nonlinear interactions leads to a transfer of energy toward small vertical scales and lower frequencies: the result is a steep energy decrease due to wave breaking. Induced mixing is evaluated. A parametric forcing is also introduced in order to compare with laboratory experiments. Wave breaking also occurs but as opposed to the unforced case different phases are next observed: internal wave growth due to constructive forcing alternate with energy decrease.     

Gravitational and acoustic waves due to the motion of an oscillating thin plate beneath the surface of a heavy weakly-compressible fluid

The three-dimensional problem of the motion of a thin plate in an inviscid, heavy, weakly-compressible fluid is solved. The surface tension is disregarded. The plate moves rectilinearly at a constant velocity under the surface of an infinite-depth fluid and oscillates at a given frequency. The fluctuating dipole potential is obtained from the Euler and continuity equations with account for the conditions on the free surface (linear theory of small waves) and the conditions at infinity. The density distribution function of the dipole layer is determined from the boundary conditions imposed on the plate surface. Formulas for calculating the far acoustic field are derived. The calculations for a square plate are carried out.     

Numerical simulation of plane and spatial nonlinear stationary waves in a two-layer fluid of arbitrary depth

The solution of a model differential equation for the three-dimensional perturbations of the interface between two immiscible fluids of different densities lying between a stationary nondeformable bottom and cover is presented. It is assumed that the waves have an arbitrary length and small, though finite, amplitude. The shapes of stationary traveling internal waves, both periodic in the two horizontal coordinates and soliton-like, are presented. These shapes depend on different parameters of the problem: the direction of the perturbation wave vector and the fluid layer depth and density ratios.     

Surface effects of internal wave generated by a moving source in a two-layer fluid of finite depth

IntroductionInternalwaves,whichcanbeexcitedbymanykindsofdisturbancesfromthesurface,bottomorinteriorofastratifiedocean ,suchasthewindstrain ,theflowoverunevenbottomsandthemovingbodyatthesurfaceorunderwater,areremarkablyobviousasthereexistsapycnoclineorathermoclineintheocean[1,2 ] .ThepropagationofgravitysurfacewaveindeepwatergeneratedbyasteadilymovingdisturbanceisrestrictedinaV_shapedregionwithahalfangleθc =19.5°,whichiswell_knownKelvinshipwave[3 ] .Thewaveinducedbyamovingunderwaterobjectisa…