Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

Internal wave generation by a fountain in a stratified fluid

The purpose of the study is the direct numerical and theoretical modeling of fountain dynamics in a fluid with density stratification in the form of a pycnocline. The fountain is formed as a vertical jet penetrates through the pycnocline. In numerical simulation the jet flow is initiated by means of preassigning a boundary condition in the form of an upward-directed laminar flow of a neutral-buoyancy fluid with an axisymmetric Gaussian velocity profile. The calculations show that at a Froude number Fr greater than a certain critical value the flow becomes unstable and the fountain executes self-oscillations accompanied by internal wave generation in the pycnocline. Depending on Fr, two self-oscillation modes can be distinguished. At fairly low Fr the fountain executes circular motion in the horizontal plane, in the vicinity of the center of jet, its shape remaining almost invariant. In this case, internal waves in the form of unwinding spirals are radiated. At fairly high Fr another mode predominates, when the fountain top chaotically “strays” in the vicinity of the center of the jet and, periodically breaking down, generates wave packets propagating toward the periphery of the computation domain. In both cases, the main peak in the frequency spectrum of the internal waves coincides with the fountain top oscillation frequency which monotonically decreases with increase in Fr. In numerical simulation the Fr-dependence of the fountain top oscillation amplitude is in good agreement with that predicted by the theoretical model of the concurrence of the interacting modes in the soft self-excitation regime.     

内孤立波作用下潜深对潜体运动响应和载荷特性的影响研究

内孤立波常发生于海洋密度跃层, 因其峰高谷深、携带能量巨大, 在传播过程中会导致跃层上下的海水流动呈现剪切状态, 并引起突发性的强流. 潜体在水下悬停时极有可能会遭遇内孤立波, 由于内孤立波的流场特性, 置于跃层上下的悬浮潜体所产生运动响应和水动力载荷变化不尽相同, 甚者会出现掉深现象. 为探究潜深对波体耦合作用的影响, 基于不可压缩N-S方程和mKdV理论, 采用速度入口造波, 结合重叠网格技术和流固耦合方法, 建立了分层流中内孤立波耦合水下潜体多自由度运动的数值模型, 通过该模型分析了不同潜深下悬浮潜体的运动响应和载荷特性. 结果表明: 在内孤立波作用下, 位于密度跃层上方和跃层中的潜体顺着波的前进方向运动, 先下沉后抬升, 位于跃层下方的潜体则会逆流持续下沉; 潜体与波面的垂向距离越小, 对其纵荡、垂荡和速度的影响越显著, 而位于密度跃层中的潜体在分界面处沿着波形运动, 其运动响应和载荷变化受影响较小; 潜体在跃层上、下流体中所受水平力的方向相反, 水平力峰值小于垂向力峰值, 且位于跃层下方的潜体一直受到低头力矩, 最终导致掉深.      

Experimental Investigation of the Generation of Internal Waves by a Vertical Cylinder in a Near-Surface Pycnocline

A bench study of the amplitudes, mode composition, and phase structure of the internal waves generated by a vertical cylinder in the presence of a near-surface pycnocline has been performed; the pycnocline took the form of a stratified fluid layer located between two quasi-homogeneous layers of thicknesses h ₁ and h ₂=2h ₁. In the experiments, the cylinder traveled at velocities critical with respect to internal wave generation. Different cases of model submergence relative to the pycnocline are considered. The dependence of the mode structure and the amplitude-phase characteristics of the forced internal waves on the body velocity and its relative submergence is analyzed. The parameters of both steady and unsteady wave systems are studied.The data obtained make it possible to predict the forced wave parameters and the critical body velocities for given model dimensions and pycnocline parameters.     

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

Chaotic motion and internal resonance of forced surface waves in a water-filled circular basin

The objectives of this paper are to investigate the chaotic motions and internal resonance of nonlinear surface waves generated by a harmonic vibration applied to the side wall of a water-filled circular basin. The harmonic forcing consists of a component with a frequency near twice a fundamental resonance frequency of the basin and a smaller component with a frequency near the fundamental frequency. The amplitude equation for the excited eigenmode corresponding to the fundamental frequency is derived and the existence of chaotic motion of this equation is studied by Melnikov method. At certain critical radii of the basin, twice the fundamental frequency is also a resonance frequency and the so-called internal resonance takes place. Eigenmode corresponding to this resonance frequency can also be excited.     

Exact expressions for transient forced internal gravity waves and spatially-localized wave packets in a Boussinesq fluid

We re-examine a simple model describing the propagation of transient forced internal gravity waves in a Boussinesq fluid with constant horizontal mean velocity which was previously studied by Nadon and Campbell (Wave Motion, 2007). The waves are generated by a horizontally-periodic lower boundary condition and propagate upwards. We derive an alternative exact expression for the solution which more readily gives insight into the behaviour of the solution at high altitude. Some special cases of lower boundary conditions are considered to illustrate the features of the solution. This form of the solution allows us to use a Fourier transform to derive the solution for the more general situation where a wave packet is generated by a horizontally-localized lower boundary condition, comprising a continuous spectrum of horizontal wavenumbers or Fourier modes. This is a more realistic representation of internal gravity waves in the atmosphere and can be used as a starting point for investigating waves generated by an obstacle of finite horizontal extent such as an isolated mountain or a mountain range.     

Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal

Shear to longitudinal mode conversion via second harmonic generation is studied theoretically and computationally for plane waves in a two-dimensional, adhesive, hexagonally close-packed microscale granular medium. The model includes translational and rotational degrees of freedom, as well as normal and shear contact interactions. We consider fundamental frequency plane waves in all three linear modes, which have infinite spatial extent and travel in one of the high-symmetry crystal directions. The generated second harmonic waves are longitudinal for all cases. For the lower transverse–rotational mode, an analytical expression for the second harmonic amplitude, which is derived using a successive approximations approach, reveals the presence of particular resonant and antiresonant wave numbers, the latter of which is prohibited if rotations are not included in the model. By simulating a lattice with adhesive contact force laws, we study the effectiveness of the theoretical analysis for non-resonant, resonant, and antiresonant cases. This work is suitable for the analysis of microscale and statically compressed macroscale granular media, and should inspire future studies on nonlinear two- and three-dimensional granular systems in which interparticle shear coupling and particle rotations play a significant role.     

Diffraction of internal waves by a circular cylinder near the pycnocline

Propagation of internal waves over a circular cylinder under the conditions of a continuous stratification characterized by the presence of a high-gradient density layer (the pycnocline) of finite thickness is studied. The dependences of the coefficent of wave propagation on the wavelength of the first-mode incident wave for various thicknesses of the pycnocline are obtained. In the diffraction of internal waves, substantial nonlinear effects are shown to occur, which result in the appearance of waves of double oscillation frequency compared to the frequency of the incident waves. The generation coefficient for these waves is found. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 79–85, March–April, 1999.     

Linear stability of the Couette flow of a vibrationally excited gas. 2. viscous problem

Based on the linear theory, stability of viscous disturbances in a supersonic plane Couette flow of a vibrationally excited gas described by a system of linearized equations of two-temperature gas dynamics including shear and bulk viscosity is studied. It is demonstrated that two sets are identified in the spectrum of the problem of stability of plane waves, similar to the case of a perfect gas. One set consists of viscous acoustic modes, which asymptotically converge to even and odd inviscid acoustic modes at high Reynolds numbers. The eigenvalues from the other set have no asymptotic relationship with the inviscid problem and are characterized by large damping decrements. Two most unstable viscous acoustic modes (I and II) are identified; the limits of these modes were considered previously in the inviscid approximation. It is shown that there are domains in the space of parameters for both modes, where the presence of viscosity induces appreciable destabilization of the flow. Moreover, the growth rates of disturbances are appreciably greater than the corresponding values for the inviscid flow, while thermal excitation in the entire considered range of parameters increases the stability of the viscous flow. For a vibrationally excited gas, the critical Reynolds number as a function of the thermal nonequilibrium degree is found to be greater by 12% than for a perfect gas.     

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.     

Diffuse mode bifurcation of soil causing convection-like shear investigated by group-theoretic image analysis

Diffuse mode bifurcation of soil under plane-strain compression test is shown, by means of an image analysis based on group-theoretic bifurcation theory, to trigger convection-like shear and to precede shear band formation. First digital photos of Toyoura sand specimens are processed by PIV (particle image velocimetry) to gather digitized images of deformation. Next bifurcation from a uniform state is detected by expanding these images into the double Fourier series and finding a predominant harmonic diffuse bifurcation mode based on that theory. This harmonic bifurcation mode, which is the mixture of a few harmonic functions, expresses complex convection-like shear. Last bifurcation from a non-uniform state is detected by decomposing each image into a few images with different symmetries to extract non-harmonic diffuse bifurcation modes. Diffuse modes of bifurcation, which hitherto were hidden behind predominant uniform compressive deformation, have thus been made transparent by virtue of the group-theoretic image analysis proposed. A possible course of deformation suggested herein is the evolution of diffuse mode bifurcation with a convection-like bifurcation mode breaking uniformity and symmetry, followed by the formation of shear bands through localization.     

Instability waves and low-frequency noise radiation in the subsonic chevron jet

Spatial instability waves associated with lowfrequency noise radiation at shallow polar angles in the chevron jet are investigated and are compared to the round counterpart. The Reynolds-averaged Navier–Stokes equations are solved to obtain the mean flow fields, which serve as the baseflow for linear stability analysis. The chevron jet has more complicated instability waves than the round jet, where three types of instability modes are identified in the vicinity of the nozzle, corresponding to radial shear, azimuthal shear,and their integrated effect of the baseflow, respectively. The most unstable frequency of all chevron modes and round modes in both jets decrease as the axial location moves downstream. Besides, the azimuthal shear effect related modes are more unstable than radial shear effect related modes at low frequencies. Compared to a round jet, a chevron jet reduces the growth rate of the most unstable modes at downstream locations. Moreover, linearized Euler equations are employed to obtain the beam pattern of pressure generated by spatially evolving instability waves at a dominant low frequency St = 0.3, and the acoustic efficiencies of these linear wavepackets are evaluated for both jets. It is found that the acoustic efficiency of linear wavepacket is able to be reduced greatly in the chevron jet, compared to the round jet.     

Vortex structure simulation for supersonic mixing layers using nonlinear PSE method

The method of nonlinear parabolized stability equations (PSE) is applied in the simulation of vortex structures in compressible mixing layer. The spatially-evolving unstable waves, which dominate the vortex structure, are investigated through spatial marching method. The instantaneous flow field is obtained by adding the harmonic waves to basic flow. The results show that T-S waves do not keep growing exponentially as the linear evolution, the energy transfer to high order harmonic modes, and that finally all harmonic modes get saturated due to nonlinear interaction. The mean flow distortion induced by the nonlinear interaction between the harmonic modes and their conjugate harmonic ones, makes great change of the average flow and increases the thickness of mixing layer. PSE methods can well capture the two- and three-dimensional large scale nonlinear vortex structures in mixing layers such as vortex roll-up, vortex pairing, and Λ vortex.     

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.     

Two-component hot-film probe for measurements of wall shear stress

A two-component, hot-film probe has been developed, to measure the wall shear stress as a vector quantity in time-dependent flows. The probe has been applied for the measurements of the bottom shear stress in a flow generated by combined waves and current in a water basin where the magnitude and the direction of the bottom shear stress varied periodically with respect to time. The probe has also been used to measure the bottom shear stress around a vertical cylinder exposed to water waves generated in a wave flume.     

Nonlinear dynamics of hydrostatic internal gravity waves

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.      

Combination Internal Resonances in Heated Annular Plates

Nonlinear modal interactions in the forced vibrations of a thermally loaded pre-buckled annular plate with clamped–clamped immovable boundary conditions are investigated. The mechanism responsible for the interaction is a combination internal resonance involving the natural frequencies of the three lowest axisymmetric modes. The in-plane thermal load acting on the plate is assumed to be axisymmetric and the plate is externally excited by a harmonic force. The nonlinear von Kármán plate equations along with the heat conduction equation are combined to model the behavior of the system. An analytical/numerical approach is used to examine the plate vibrations to a harmonic excitation near primary resonance of one of the modes.     

Parametrically Excited Non-Linear Traveling Beams with and without External Forcing

The effects of parametric excitation on a traveling beam, both with and without an external harmonic excitation, have been studied including the non-linear terms. Non-linear, complex normal modes have been used for the response analysis. Detailed numerical results are presented to show the effects of non-linearity on the stability of the parametrically excited system. In the presence of both parametric and external harmonic excitations, the response characteristics are found to be similar to that of a Duffing oscillator. The results are sensitive to the relative strengths of and the phase difference between the two forms of excitations.     

Flow pattern behind a submerged body near a pycnocline

A flow past an ellipsoid immersed in a flow of a viscous stratified fluid is studied using a Large Eddy Simulation (LES) method for different locations of the body relative to a density discontinuity. It is shown that, even when the internal Froude number is large, for small angles of attack of the body the stratification affects its drag force. When the body is located above the pycnocline, the presence of the discontinuity results in the ascending of a vortex filament. A spectral analysis showed that the internal waves in the body wake have a multimode structure.