Derivation of a higher order nonlinear Schrödinger equation for weakly nonlinear Rossby waves

In the paper, with the help of a perturbation expansion method a new higher order nonlinear Schrödinger (HNLS) equation is derived to describe nonlinear modulated Rossby waves in the geophysical fluid. Using this equation, the modulational wave trains are discussed. It is found that the higher order terms favor the instability growth of modulational disturbances superimposed on uniform Rossby wave trains, but the instability region becomes narrower. In addition, the latitude and uniform background basic flow are found to affect the instability growth rate and instability region of uniform Rossby wave train. However, for a geostrophic flow the background basic flow does not affect the modulational instability of uniform Rossby wave train.     

Non-conservative pressure-based compressible formulation for multiphase flows with heat and mass transfer

A pressure-based compressible multiphase flow solver has been developed based on non-conservative discretization of the mixture continuity equation. The formulation is an extension of the single phase incompressible pressure-correction approach, such that it can be applied to both two-phase flows using interface resolving methods and general n-phase ensemble-averaged mixture flows. The formulation is currently presented with the single pressure and single temperature assumption, but extension to multiple temperatures is straightforward. A robust treatment of phase change allows the method to model conditions with rapid phase change such as expansion through nozzles and valves. The method has been validated thoroughly using canonical single phase problems such as the shock tube, tank filling and sudden valve closure problems. Multiphase flow validation has been carried out for sound propagation in mixtures using the ensemble-averaged model and pressure wave transmission and reflection across an air-water interface, using the level set interface tracking method. The method has been used to study sound propagation in saturated steam-water systems under thermodynamic non-equilibrium, where the expected drastic reduction in the speed of sound is reproduced. Finally the method is applied to the problem of critical (choked) flow in a nozzle for a saturated steam-water system.     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.     

垂直激励圆柱形容器中的表面波特性研究

利用奇异摄动理论的两时间变量展开法，研究了垂直强迫激励圆柱形容器中的单一水表面驻波模式。假设流体是无粘、不可压且运动是无旋的，在忽略了表面张力的影响下，得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算，并研究了特定(3，4)模式的表面驻波结构和特性，如驻波的节点分布及随某些参数的变化规律等，从计算的等高线的图象来看，和以往的实验结果相当吻合。     

有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

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.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

EFFECTS OF VARYING BOTTOM ON NONLINEAR SURFACE WAVES

The resonant flow of an incompressible, inviscid fluid with surface tension on varying bottoms was researched. The effects of different bottoms on the nonlinear surface waves were analyzed. The waterfall plots of the wave were drawn with Matlab according to the numerical simulation of the fKdV equation with the pseudo-spectral method. Prom the waterfall plots, the results are obtained as follows: for the convex bottom, the waves system can be viewed as a combination of the effects of forward-step forcing and backward step forcing, and these two wave systems respectively radiate upstream and downstream without mutual interaction. Nevertheless, the result for the concave bottom is contrary to the convex one. For some combined bottoms, the wave systems can be considered as the combination of positive forcing and negative forcing.     

Three-dimensional divergent waves on a model viscoelastic coating in a potential incompressible fluid flow

Generation of three-dimensional nonlinear waves on a model viscoelastic coating in a potential flow of an incompressible fluid is studied. Periodic nonlinear waves enhanced by the development of quasi-static instability (wave divergence) are considered. The coating is modeled by a flexible flat plate supported by a distributed nonlinearly-elastic spring foundation. Plate flexure is described on the basis of the Karman equations of the theory of thin plates. Perturbations of surface pressure in the potential flow are found in the small slope approximation to an accuracy to terms of the second order of smallness. Numerical simulation reveals a jump-like transition from two-dimensional nonlinear waves to three-dimensional wave structures, which are also observed in experiments.     

Radiation and scattering of sound from a vortex ring

The mechanisms of generation and scattering of sound by a vortex ring are investigated on the basis of fluid dynamics. The vortex ring can serve as a simple dynamic model of the large-scale structures observed in shear flows. Moreover, it is probably the most easily studied vortex element that can be created experimentally. The sound scattering investigation also served to determine the extent to which the vortex is affected by sound, its selectivity with respect to such parameters as the acoustic frequency, the angle of incidence of the wave, etc. The perturbed motion is considered against the background of the steady-state motion of the ring. The perturbed motion in the vortex core is determined on the basis of linear incompressible fluid dynamics. Two terms of the expansion in the M number of the far acoustic field generated by the perturbations in the core are found in accordance with Lighthill's theory. The acoustic power and directivity of the radiation and the acoustic instability growth rate are calculated. It is shown that the scattering of sound by the vortex ring is a resonance effect, and the scattering amplitude near resonance is determined. The acoustic action on the hydrodynamic structure of the flow in the core of the ring is especially intense near the resonances and extends over a period short as compared with the characteristic time of the acoustic instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 83–95, May–June, 1987.     

SOLITARY WAVES IN FINITE DEFORMATION ELASTIC CIRCULAR ROD

A new nonlinear wave equation of a finite deformation elastic circular rod simultaneously introducing transverse inertia and shearing strain was derived by means of Hamilton principle. The nonlinear equation includes two nonlinear terms caused by finite deformation and double geometric dispersion effects caused by transverse inertia and transverse shearing strain. Nonlinear wave equation and corresponding truncated nonlinear wave equation were solved by the hyperbolic secant function finite expansion method. The solitary wave solutions of these nonlinear equations were obtained. The necessary condition of these solutions existence was given also.     

Free surface flow under gravity and surface tension due to an applied pressure distribution: I Bond number greater than one-third

We consider steady free surface two-dimensional flow due to a localized applied pressure distribution under the effects of both gravity and surface tension in water of constant depth, and in the presence of a uniform stream. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The behavior of the forced nonlinear waves is characterized by three parameters: the Froude number, F, the Bond number, τ > 1/3, and the magnitude and sign of the pressure forcing parameter ɛ. The fully nonlinear wave problem is solved numerically by using a boundary integral method. For small amplitude waves and F < 1 but not too close to 1, linear theory gives a good prediction for the numerical solution of the nonlinear problem in the case of bifurcation from the uniform flow. As F approaches 1, the nonlinear terms need to be taken account of. In this case the forced Korteweg-de Vries equation is found to be an appropriate model to describe bifurcations from an unforced solitary wave. In general, it is found that for given values of F < 1 and τ > 1/3, there exists both elevation and depression waves. In some cases, a limiting configuration in the form of a trapped bubble occurs in the depression wave solutions.     

A lattice Boltzmann model for two‐dimensional sound wave in the small perturbation compressible flows

A multi‐entropy‐level lattice Boltzmann model for two‐dimensional sound wave equation in the small perturbation flows is presented. In this model, we used higher‐order moment method, multi‐scale technique and the Chapman–Enskog expansion, and multi‐entropy‐level to obtain sound wave equation with isentropic equation. As numerical examples, the Doppler effects in the sound wave propagation, the sound scattering from circular cylinder are simulated. The numerical results show that this model can be used to simulate sound wave propagation. Copyright © 2010 John Wiley & Sons, Ltd.     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

Analysis of heating a three-dimensional porous bed utilizing the two energy equation model

In this paper heating a three-dimensional porous packed bed by a non-thermal equilibrium flow of incompressible fluid is analytically investigated. A two energy equation model is employed to simulate the temperature difference between the fluid and solid phases. Using the perturbation technique, an analytical solution for the problem is obtained. It is shown that the temperature difference between the fluid and solid phases forms a wave localized in space and propagating from the fluid inlet boundary in the direction of the flow. The amplitude of the wave decreases while the wave propagates downstream.     

Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum

The solutions of initial and boundary value problems of the outflow of an ideal (inviscid and non-heat-conducting) gas from cylindrical and spherical sources into a vacuum are obtained. Time is measured from the moment, when the source is turned on; at this moment the source is surrounded by a vacuum. The entropy, flow rate, and the Mach number of the gas outflowing from the source are given, together with the source radius; the Mach number can be greater of or equal to unity. If the source radius is greater than zero, then the flow domain in the “radial coordinate–time” plane consists of the stationary source flow and adjoining non-self-similar centered expansion wave consisting of C^?-characteristics. The stationary flow is described by the known formulas, while the expansion wave is calculated by the method of characteristics. The calculations by this method confirm the earlier obtained laws for large values of the radial coordinate. The interface between the vacuum and the expansion wave is the straight trajectory of particles and, at the same time, a unique rectilinear C^?-characteristic. For the source of zero radius (“pointwise” source) the velocity, density, and speed of sound of the outflowing gas are infinite. The gas velocity remains infinite everywhere, while the density and speed of sound become zero for any non-zero values of the radial coordinate. For the pointwise source the problem of outflow into a vacuum is self-similar. In the plane of the “self-similar” velocity and speed of sound its solution is given by three singular points of a differential equation in these variables. At one of these points the self-similar velocity is infinite, the self-similar speed of sound is zero, and the self-similar independent variable varies from zero to infinity, with the exception of the extreme values.     

Asymptotic of the flow upon shock incidence on a wedge cavity

The asymptotic of the motion originating because of shock incidence on a wedge cavity in a metal is investigated as the wave amplitude tends to zero. It has been shown in [1] that the flow is hence divided into two domains. The principal term governing the flow in the first domain agrees with the acoustic approximation. The flow in the second domain is described by incompressible fluid equations in the principal term. Determination of the flow in the second domain is reduced herein to the solution of a singular nonlinear integral equation. A numerical solution is found for a series of values of the cavity aperture.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 129–138, May–June, 1972.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性,同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应,利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析,在不同的条件下,该方程在相平面上存在同宿轨道或异宿轨道, 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法,对该非线性方程进行 求解,得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解,讨论了这些解存 在的必要条件,这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程,从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

Stability of wave forms of motion of a viscous fluid layer under tangential stress

We study the stability of wave flow of a viscous incompressible fluid layer subjected to tangential stress and an inclined gravity force with respect to long-wave disturbances.An asymptotic solution is constructed for the equations of the disturbed motion and the problem is reduced to the study of a second-order ordinary differential equation. It is shown that after loss of stability by a Poiseuille flow the laminar nature of the flow is not destroyed, but the form of the free surface acquires a wave-like profile. The Poiseuille regime is stable for low Reynolds numbers. The critical Reynolds number for wave flow is found, and the stability and instability regions are determined.