Body wave propagation in rotating elastic media

We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation Ω. Wave propagation is shown to be directly related to the Kibel number Ki=ω/Ω, where ω is the wave frequency. Two dispersive waves W₁ and W₂ are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W₁ shows a cutoff frequency ω_c=Ω below which it does not propagate. The case of small angular rotation Ω is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order O(Ki⁻¹).     

Dynamics of Rossby solitary waves with time-dependent mean flow via Euler eigenvalue model

The investigation on the fluctuations of nonlinear Rossby waves is of great importance for the understanding of atmospheric or oceanic motions. The present paper mainly deals with the well-known atmospheric blocking phenomena through the nonlinear Rossby wave theories and the corresponding methods. Based on the equivalent barotropic potential vorticity model in the β-plane approximation underlying a weak time-dependent mean flow, the multiscale technique and perturbation approximated methods are adopted to derive a new forced Korteweg-de Vries model equation with varied coefficients (vfKdV) for the Rossby wave amplitude. For a further analytical treatment of the obtained model problem, a special kind of basic flow is adopted. The evolution processes of atmospheric blocking are well discussed according to the given parameters according to the dipole blocking theory. The effects of some physical factors, especially the mean flow, on the propagation of atmospheric blocking are analyzed.

     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

An experimental investigation of roll waves in high pressure two-phase inclined pipe flows

This paper examines the effects of small upward inclinations on the formation of roll waves and the properties of fully developed roll waves at high pressure conditions. A total of 984 experiments were conducted at six positive pipe inclinations θ = 0.00°, 0.10°, 0.25°, 1.00°, 2.50° and 5.00° using a 25 m long 10 cm i.d. pipe. Sulfur hexafluoride (SF₆) was used at 8 bara giving a gas density of 50 kg/m³. Two independent mechanisms for the formation of roll waves were identified; (1) interaction between 2D shallow water waves and (2) a visible long wavelength instability of the stratified layer. Viscous long wavelength linear stability analysis predicted the critical liquid flow rate and liquid height for the initiation of roll waves when roll waves were formed due to the second mechanism. A simple equation from shallow water wave theory agreed with measurements for critical liquid flow rate when roll waves were formed due to the first mechanism. Shallow water wave speed agreed with critical wave speeds at transition and nonlinear wave speeds for fully developed roll waves in certain cases. The increase in interfacial friction due to the presence of large waves was compared with models from the literature.     

Non-linear water waves on shearing flows

This article is devoted to the study of the propagations of the nonlinear water waves on the shear flows. Assuming μ=kh is small andε/μ ²∼O(1), and the base flow is uniformly sheared, the modified Boussinesq equation is obtained. We calculate propagations of the single solitary wave with vorticity Γ=0,>0 and <0. The influences of the vorticity are manifested. At the end examples of the interactions of two solitary waves, moving in opposite and the same directions, are given. Besides the phase shift, there also occur second wavelets after head-on collision. The project supported by the National Natural Science Foundation of China     

Theory of weak turbulence of waves on a fluid surface

The nonlinear interaction of waves in a fluid of finite depth is discussed. Forbidden decay processes in the gravitational portion of the spectrum are eliminated from the Hamiltonian by means of a canonical transformation. This provides an opportunity to obtain a kinetic equation which takes into account scattering of capillary waves by gravitational waves, in addition to decays in the subsystem of gravitational waves. The distribution N_k P^1/2h^1/4k^–4 is obtained for capillary waves in shallow water with constant flow of energy P with respect to the spectrum in the space of the wave numbers k. The interaction of the gravitational and capillary turbulence spectra is discussed. An induced distribution of gravitational waves is found which results from their interaction with capillary waves. It is an increasing function of the wave numbers q in the region bounded by the capillary constant k_o, N_q q^9/4 (q < k_o). The coupling of spectra in the gravitational and capillary regions and the conversion from slightly turbulent distributions to universal distributions are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 97–106, November–December, 1974.     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

Postcritical regimes in the nonlinear problem of vortex motion under the free surface of a weighable fluid

An improved Levi-Civita method in which the singularities of the desired function are taken into account by introducing terms containing power singularities is proposed. Results of numerical analysis of the nonlinear problem of a vortex in a bounded flow of an ideal weighable fluid (Fr>1) are given. The following limiting flow regimes are studied: the Stokes waves with one and two crests, emergence of a critical point on the surface, and the detachment of a vortex from a soliton and a uniform flow. It is shown that nonperiodic waves can form in a local zone in the vicinity of the critical point. Ufa State Technical Aviation University, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 70–76, January–February, 2000.     

A comparative study of two fast nonlinear free‐surface water wave models

This paper presents a comparison in terms of accuracy and efficiency between two fully nonlinear potential flow solvers for the solution of gravity wave propagation. One model is based on the high‐order spectral (HOS) method, whereas the second model is the high‐order finite difference model OceanWave3D. Although both models solve the nonlinear potential flow problem, they make use of two different approaches. The HOS model uses a modal expansion in the vertical direction to collapse the numerical solution to the two‐dimensional horizontal plane. On the other hand, the finite difference model simply directly solves the three‐dimensional problem. Both models have been well validated on standard test cases and shown to exhibit attractive convergence properties and an optimal scaling of the computational effort with increasing problem size. These two models are compared for solution of a typical problem: propagation of highly nonlinear periodic waves on a finite constant‐depth domain. The HOS model is found to be more efficient than OceanWave3D with a difference dependent on the level of accuracy needed as well as the wave steepness. Also, the higher the order of the finite difference schemes used in OceanWave3D, the closer the results come to the HOS model. Copyright © 2011 John Wiley & Sons, Ltd.     

Waves Ahead of an Underwater Wing. An Experiment

The results of an experimental study of nonlinear waves, both smooth and breaking, formed during the motion of a symmetric wing in a channel with a plane horizontal bottom are presented. It is found that if the wing velocity U ₀ is close to the limiting speed c ₂ of propagation of free solitary waves, then the following qualitative changes in the flow pattern take place. For U ₀= c ₂ and a strictly specified combination of the other relevant parameters, a forced solitary wave of limiting amplitude is formed above the wing. Behind the wing level depression gives way to elevation. If the blocking ratio is not too large, the waves do not propagate upstream for U ₀ > c ₂. If the blocking ratio is large, the disturbances can propagate far upstream even for U ₀ > c ₂; however, in this case the leading front of the disturbances breaks.     

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

Design of a tunnel-entrance hood with multiple windows and variable cross-section

A theory is proposed for the design of a uniform tunnel-entrance hood whose cross-sectional area exceeds the tunnel area . A train entering the tunnel produces a low-frequency compression wave that can be subject to nonlinear steepening in a long tunnel. An optimized hood of length ℓ_h increases the initial thickness of the compression wave front from R/M to ℓ_h/M, where Rℓ_h is the nominal radius of the tunnel and M is the train Mach number. In addition, the pressure rise should be linear across the wave front to obtain an overall minimum value of the subjectively important pressure gradient. This is achieved in a uniform hood by distributing windows along the hood wall to vent away high-pressure air displaced by the train. We consider the problem of determining the distribution and sizes of these windows and the magnitude of the area ratio to ensure that the hood behaves optimally at low-train Mach numbers (M<0.2), when the hood can be regarded as being acoustically compact. At the projected higher Mach numbers of advanced high-speed trains (0.4, say) recent analysis for hoods of uniform cross-section by Howe in 2003 indicates that a hood optimized for low Mach number operations continues to produce an essentially linear pressure rise across a compression wave of thickness ℓ_h/M except for a low-amplitude oscillation at the very front of the wave.     

Nonlinear Strong Shock Interactions: A Shock-Fitted Approach

This paper addresses nonlinear effects which result from the interaction of shock waves with vortices. A series of experiments are carried out, which involve the interaction of a strong shock wave with a single plane vorticity wave and a randomly distributed wave system. These experiments are first conducted in the linear regime to obtain a mutual verification of theory and computation. They are subsequently extended into the nonlinear regime. A systematic study of the interaction of a plane shock wave and a single vortex is then conducted. Specifically, we investigate the conditions under which nonlinear effects become important, both as a function of shock Mach number, M ₁, and incident vortex strength (characterized by its circulation Γ). The shock Mach number is varied from 2 to 8, while the circulation of the vortex is varied from infinitesimally small values (linear theory) to unity. Budgets of vorticity, dilatation, and pressure are obtained. They indicate that nonlinear effects become more significant as both the shock Mach number and the circulation increase. For Mach numbers equal to 5 and above, the dilatation in the vortex core grows quadratically with circulation. An acoustic wave propagates radially outward from the vortex center. As circulation increases, its upstream-facing front steepens at low Mach numbers, and its downstream-facing front steepens at high Mach numbers. A high Mach number asymptotic expansion of the Rankine--Hugoniot conditions reveals that nonlinear effects dominate both the shock motion and the downstream flow for ΓM ₁ > 1. Received 28 June 1997 and accepted 25 November 1997     

The Benjamin–Lighthill Conjecture for Near-Critical Values of Bernoulli’s Constant

In 1954, Benjamin and Lighthill made a conjecture concerning the classical nonlinear problem of steady gravity waves on water of finite depth. According to this conjecture, a point of some cusped region on the (r, s)-plane (r and s are the non-dimensional Bernoulli’s constant and the flow force, respectively), corresponds to every steady wave motion described by the problem. Conversely, at least one steady flow corresponds to every point of the region. In the present paper, this conjecture is proved for near-critical flows (when r attains values close to one), under the assumption that the slopes of wave profiles are bounded. Another question studied here concerns the uniqueness of solutions, and it is proved that for every near-critical value of r only the following waves do exist: (i) a unique (up to translations) solitary wave; (ii) a family of Stokes waves (unique up to translations), which is parametrised by the distance from the bottom to the wave crest. The latter parameter belongs to the interval bounded below by the depth of the subcritical uniform stream and above by the distance from the bottom to the crest of solitary wave corresponding to the chosen value of r.     

Homogenization of an elasticity problem in domains with a net of slender bars near surface

We consider an elasticity problem in a domain Ω⁽⁾=ΩF⁽⁾, where Ω is an open bounded domain in R³, F⁽⁾ is a connected nonperiodic set in Ω like a net of slender bars, and is a parameter characterizing the microstructure of the domain. We consider the case of a surface distribution of the set F⁽⁾, i.e., for sufficiently small , the set F⁽⁾ is concentrated in arbitrary small neighbourhood of a surface Γ. Under a hypothesis on the asymptotic behaviour of the energy functional, we obtain the macroscopic (homogenized) model. To cite this article: M. Goncharenko, L. Pankratov, C. R. Mecanique 331 (2003).     

位势流动和非均匀介质中声波方程的36种形式

声波方程是对大多数声学问题进行数学描述的出发点. 那些得到 广泛应用的经典波动方程及对流波动方程都存在苛刻的适用条件, 即仅适用于描述处于静态或匀速运动状态的定常 均匀介质中的线性无耗散声波. 然而, 很多实际场合并不满足这些严格的适用条件. 本文对经典声波方程和对流声波 方程进行推广, 导出了编号为W1$\sim$W36的36种不同形式的声波方程, 涵盖了处于静止、势流或旋涡流状态下的非均匀 和/或非定常介质中的声波传播问题. 所考虑的声波传播情形包括: (1) 线性波, 即具有小梯度(小振幅)性质; (2)非线性波, 即具有陡峭梯度性质, 包括``波纹'(小振幅大梯度)或者大振幅波. 本文仅考虑非耗散声波, 即排除了由剪切、体积黏度及热传导所引起的耗散. 对具有匀熵或等熵(熵沿流线守恒)性质的均匀介质和非均匀介质中的声传播进行了研究但非等熵(即耗散)情况除外; 另外, 对非定常介质中的 声波问题也进行了分析. 所涉及的介质可以处于静止、匀速运动状态, 或者是非匀速的和/或非定常的平均流动, 包括: (1)低Mach数的势平均流(即不可压缩的平均态), 或高速势平均流(即非均匀可压缩的平均流); ② 变截面管 道中的准一维传播, 包括无平均流的号管和具有低或高Mach数平均流的喷管; 或③平面的、空间的、或轴对称的单 向剪切平均流. 本文没有探讨其他类型的旋涡平均流(将与耗散及其他情形一起留待下一步研究), 例如, 可能与剪切效应相结合的轴对称旋转平均流. 通过对流体力学的一般方程进行消元处理或根据声学变分原理, 导出了36种波动方程, 对一些波动方程还采用这两种方法进行相互校验. 尽管声波方程的36种形式没有涵盖非线性、非均匀与非定常及非匀速运动介质 这3个效应的所有可能的组合情形, 但它们的确包括了孤立状态下的各种效应, 并包括了多种多重效应组合的 情形. 虽然经典波动方程和对流波动方程仅适用于处于静止(或匀速运动)的均匀定常介质中的线性无耗散声波, 但它们在 相关文献中已被广泛采用; 本文给出的36种声波方程提供了它们多种有用的推广形式. 在许多实际应用中, 经典波动方 程和对流波动方程仅是粗略的近似, 声波方程的更一般形式可提供更令人满意的理论模型. 本文每节末尾给出了这些应用 的众多范例. 在这篇评论文章中引用了240篇参考文献.     

Rayleigh waves obtained by the indeterminate couple-stress theory

Rayleigh waves in a linear elastic couple-stress medium are investigated; the constitutive equations involve a length parameter l that characterizes the microstructure of the material. With , c_T=conventional transversal speed and q=wave number, an explicit expression is derived for the relation between , lq and Poisson's ratio ν. The Rayleigh speed turns out to be dispersive and always larger than the conventional Rayleigh speed. It is of interest that when lq=1 and ν≥0, it always holds that . The displacement field is investigated and it is shown that no Rayleigh wave motions exist when lq→∞ and when lq=1, ν≥0. Moreover, a principal change of the displacement field occurs when lq passes unity. The peculiarity that no Rayleigh wave motions exist when lq=1, ν≥0 may support the criticism by Eringen (1968) against the couple-stress theory adopted here as well as in much recent literature.     

Free convection heat transfer from an isothermal wavy surface in a porous enclosure

The coupled streamfuction–temperature equations governing the Darcian flow and convection process in a fluid-saturated porous enclosure with an isothermal sinusoidal bottom sun face, has been numerically analyzed using a finite element method (FEM). No restrictions have been imposed on the geometrical non-linearity arising from the parameters like wave amplitude (a), number of waves per unit length (N), wave phase (Φ), aspect ratio (A) and also on the flow driving parameter Rayleigh number (Ra). The numerical simulations for varying values of Ra bring about interesting flow features, like the transformation of a unicellular flow to a multicellular flow. Both with increasing amplitude and increasing number of waves per unit length, owing to the shift in the separation and reattachment points, a row–column pattern of multicellular flow transforms to a simple row of multicellular flow. A cycle of n celluar and n+1 cellular flows, with the flow in adjacent cells in the opposite direction, periodically manifest with phase varying between 0 and 360°. The global heat transfer into the system has been found to decrease with increasing amplitude and increasing number of waves per unit length. Only marginal changes in the global heat flux are observed, either with increasing Ra or varying Φ. Effectively, sinusoidal bottom surface undulations of the isothermal wall of a porous enclosure reduces the heat transfer into the system. © 1998 John Wiley & Sons, Ltd.     

Theoretical mean wave resistance of precursor solition generation in two-layer flow

A new concept of pseudo mean wave resistance is introduced to find theoretical mean wave resistances of the precursor soliton generation in two-layer flow over a localized topography at near-resonance in this paper. The pseudo mean wave resistance of the precursor soliton generation of two-layer flow is determined in terms of the AfKdV equation. From the theoretical results it is shown that the theoretical mean wave resistance is equal to the pseudo mean wave resistance times 1/m ₁, wherem ₁ is the coefficient of the fKdV equation. From the regional distribution of the energy of the precursor soliton generation at the resonant points, it is shown that ratios of the theoretical mean wave resistance and regional mean energy to the total mean energy are invariant constants, i.e. : (−1/2):1:1, in which and are the mean energy of the generating regions of the precursor solitons, of the depression and of the trailing, wavetrain at the resonant points respectively, and <D> are the total energy of the system and the theoretical mean wave resistance at the resonant points. A prediction of the theoretical mean wave resistance flow over the semicircular topography is carried out in terms of the theoretical results of the present paper. The comparison shows that the theoretical mean wave resistance is in good agreement with the numerical calculation. The project supported by the Foundation of The State Education Commission “The Dynamics of Upper Ocean” and the Grants of The Physical Oceanography Laboratory of Ocean University of Qingdao