Nonlinear development of instabilities in supersonic vortex sheets: I. The basic kink modes

Classical linearized stability analysis predicts (neutral) stability of supersonic vortex sheets for compressible flow with normalized Mach numbers, M > √2, while recent detailed numerical simulations by Woodward indicate the nonlinear development of instabilities for M > √2 through the development and interaction of propagating kink modes in the slip-stream. These kink modes are discontinuities in the slip-stream bracked by shock waves and rarefaction waves which grow self-similarly in time. In this paper, the apparent paradox is resolved by developing appropriate small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions which yield the response to a very small amplitude nonlinear planar sound wave incident on the vortex sheet. The analysis leads to three specific angles of incidence depending on M > √2 where nonlinear resonance occurs. For these three special resonant angles of incidence the perturbation expansions automatically yield simplified equations. These equations involve an appropriate Hamilton-Jacobi equation for the perturbed vortex sheet location; the derivative of the solution of this Hamilton-Jacobi equation provides boundary data for two nonlinear Burgers transport equations for the sound wave emanating from the two sides of the vortex sheet. These equations are readily solved exactly and lead to the quantitative time-dependent nonlinear development of three different types of kink modes with a structure similar to that observed by Woodward.     

水下涡流场前向声散射特性分析

研究水下涡声散射特性,在目标探测和流场声成像领域具有重要意义。针对水下低马赫数涡流场前向声散射建立了数值计算方法,探究了其形态函数和指向性。首先,基于摄动声学理论给出了考虑流声耦合作用的涡声散射模型,采用时域有限差分结合完美匹配层构建了数值求解方法;随后,在算法验证的基础上,预报分析了高斯涡涡核尺寸在1~10 m,同时入射平面波无量纲波数在1~10范围内,涡流场强度对前向声散射特性的影响。结果表明,低马赫数下,声散射场具有对称性,且有明显的主瓣和指向性。其前向散射形态函数随入射波波数、涡核尺寸、涡流场强度增加而增大;主瓣方位角随波数增加而趋近入射波传播方向。      

Second-harmonic pressure generation of a non-diffracting acoustical high-order Bessel vortex beam of fractional type α

Background and motivation

Generalized Bessel vortex beams are regaining interest from the standpoint of acoustic scattering and radiation force theories for applications in particle rotation, mixing and manipulation. Other possible applications may include medical and nondestructive imaging. To manipulate heavy particles in a host medium, a minimum threshold of the incident sound field intensity is required at relatively high wave amplitudes such that nonlinear wave propagation occurs and the generation of harmonics may be detected. Thus, predictions of the harmonics generation become crucial from the standpoint of experimental design, and the present analysis should assist in the development of more complete models related to the (nonlinear) scattering and radiation forces under such circumstances. The purpose of this research is to construct a theoretical model for the second-harmonic pressure generation associated with a category of non-diffracting Bessel vortex beams known as high-order Bessel vortex beams of fractional typeα (HOBVBs-Fα).

Method

The weakly nonlinear wave propagation of a HOBVB-Fα is investigated based on Lighthill’s formalism. Analytical solutions up to the second-order level of approximation are derived and discussed. Closed-form solutions are obtained, which are expressed as a function of first-order quantities available from the classical linear theory. Lateral profiles of the HOBVB-Fα are computed and compared.

Results and conclusion

The results show that the beam’s width reduces and becomes narrower, the side-lobes decrease in magnitude, and the hollow region diameter (or null in magnitude) increases as the order of nonlinearity increases. Furthermore, the nonlinearity of the medium preserves the non-diffracting feature of the beam’s second-harmonic generation within the pre-shock range.     

Investigation of scattering processes in quantum few-body systems involving long-range interaction by the complex-rotation method

The complex-rotation method adapted to solving the multichannel scattering problem in the two-body system where the interaction potential contains the long-range Coulomb components is described. The scattering problem is reformulated as the problem of solving a nonhomogeneous Schrödinger equation in which the nonhomogeneous term involves a Coulomb potential cut off at large distances. The incident wave appearing in the nonhomogeneous term is a solution of the Schrödinger equation with longrange Coulomb interaction. This formulation is free from approximations associated with a direct cutoff of Coulomb interaction at large distances. The efficiency of this formalism is demonstrated by considering the example of solving scattering problems in the α-α and p-p systems.     

On the convergence of the amplitude of the diffracted nonstationary wave in scattering by wedges

We make more precise the Limiting Amplitude Principle in the two-dimensional scattering of an incident plane harmonic wave by a wedge. We find the long-time asymptotic regime of convergence of the amplitude of the cylindrical wave diffracted by the vertex of a wedge to the limiting amplitude of the solution to the corresponding stationary problem. The asymptotics turns out to be uniform on compacta and depends on the magnitude of the wedge and the profile of the incident wave. The cases of Dirichlet-Dirichlet and Dirichlet-Neumann boundary conditions are considered.     

Irregular scattering of acoustic rays by vortices

The scattering of high-frequency sound wave, under geometrical acoustic approximation, by three stationary vortices in two dimensions is investigated. For a sufficiently high Mach number of the vortex flow, the scattering of sound rays becomes irregular, displaying a new example of chaotic scattering for a time-reversal breaking system. The fractal dimension, as well as the unstable and stable manifolds of the scattering dynamics, is presented.     

Dynamics of D’Alembert wave and soliton molecule for a (2+1)-dimensional generalized breaking soliton equation

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.     

Alpha-Be asymptotic wave function as three-α particle scattering

The asymptotic form of the wave function for scattering from an initial state of α-particle incident on ⁸Be, considered as two bound α-particles, with enough energy to give a final state of three free α-particles is given. Our resulted equation with the obtained expression is suitable for numerical calculations. Applying this equation to the analysis of experimental data, very good agreement is obtained.     

Considerations to Rayleigh’s hypothesis

In 1907 Lord Rayleigh published a paper on the dynamic theory of gratings. In this paper he presented a rigorous approach for solving plane wave scattering on periodic surfaces. Moreover he derived explicit expressions for a perfectly conducting sinusoidal surface, and for perpendicular incidence of the electromagnetic plane wave. This paper was criticized by Lippmann in 1953 for he assumed Rayleigh’s approach to be incomplete. Since this time there have been published several arguments, proofs, and discussions concerning the correctness and the range of validity of Rayleigh’s approach not only for plane wave scattering on gratings but also for light scattering on nonspherical structures, in general. In the paper at hand we will discuss the different point of views on what is called “Rayleigh’s hypothesis” as well as the relevance of a found theoretical limit for its validity. Furthermore we present a numerical treatment of the original scattering problem of a p-polarized plane wave perpendicularly incident on a perfectly conducting sinusoidal surface (i.e., the scalar Dirichlet problem). In doing so we emphasizes the near-field solution especially within the grooves of the grating up to points on the surface, and below the surface. Two different Green’s function formulations of Huygens’ principle are used as starting points. One of this formulation results in the general T-matrix approach which is considered to be affected by Rayleigh’s hypothesis especially for near-field calculations. The other formulation provides a conventional boundary integral equation which is in accordance with Lippmann’s point of view and free of problems with Rayleigh’s hypothesis. But the obtained results show that Lippmann’s argumentation do not withstand a critical numerical analysis, and that the independence of least-squares approaches from Rayleigh’s hypothesis, as understood and proven by Millar, seems to hold also for certain methods which does not fit into such an approach.     

Scattering and active acoustic control from a submerged piezoelectric-coupled orthotropic hollow cylinder

A general exact analysis for three-dimensional scattering of a time-harmonic plane-progressive sound wave obliquely incident upon an arbitrarily thick bilaminated circular hollow cylinder of infinite extent, which is composed of a cylindrically orthotropic axially polarized piezoelectric inner layer perfectly bonded to an orthotropic outer layer, is presented. An approximate laminate model in the context of the so-called state space formulation along with the classical T-matrix solution technique involving a system global transfer matrix is employed to solve for the unknown modal scattering and transmission coefficients. Numerical example is given for an air-filled and water-submerged two-layered elasto-piezoelectric hybrid (steel/PZT4) hollow cylinder insonified by an obliquely incident unit-amplitude plane sound wave. Following the acoustic resonance scattering theory (RST), the total form function amplitude together with the associated global scattering, the far-field inherent background, and the resonance scattering coefficients of the nth normal mode are computed as a function of dimensionless frequency for selected angles of incidence, piezoelectric layer thickness parameters, and electrical boundary conditions (i.e., open/closed circuit or active). Also, the electrical voltage coefficients required for partial or complete cancellation of the reflected sound field are calculated. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

Electromagnetic scattering as a radiation reaction problem

Electromagnetic scattering is treated as a dynamical problem, by equating the external torque exerted by the incident wave on the sphere to the self torque due to the radiated (scattered) wave. ForI _mech=0,our scattering amplitude is equal to the usualP-wave amplitude of the electromagnetic scattering on an infinitely conducting sphere. The poles of the scattering amplitude, in particular their dependence onI _mech, are studied. For example, a pole on the positive imaginary axis, which usually corresponds to a bound state, corresponds to a runaway solution in our case. Non-decaying resonances are also discussed.     

Diffraction of a sound wave by the open end of a flanged waveguide with impedance walls

Diffraction of a plane sound wave by the open end of an impedance-wall waveguide connected to an opening in an impedance screen is considered. The plane wave is incident on the waveguide from a free half-space. Two versions of the problem are considered: for a semi-infinite waveguide and for a finite-length waveguide with a specified bottom impedance; the impedances of the walls, screen, and waveguide bottom can be different. The finite-length waveguide can be treated as an open cavity in the impedance screen. For the cavity of zero length, the problem is reduced to the diffraction by an impedance insert in the impedance screen. The solution in the external region determines the scattered field; the solution in the internal region allows one to determine the directional pattern of an array of receivers located in the cavity. The problem is solved using the integral Helmholtz equation with a specially selected Green’s function that provides the fulfillment of the boundary conditions. Formally, the problem is reduced to an infinite system of algebraic equations. The computational results obtained for bistatic and monostatic scattering patterns are presented.     

A new boundary method for electromagnetic scattering from inhomogeneous bodies

A new numerical method for scattering from inhomogeneous bodies is presented. In particular, the 2D case of a TM-polarizated incident wave scattered by an infinite cylinder is considered. The scattered field is sought in two different domains. The first one is a bounded region inside the scattering body with an inhomogeneous permittivity ε(x,y). The second one is an unbounded homogeneous region outside the scatterer. An approximate solution for the scattered field inside the scatterer is sought by applying the QTSM technique. The method of discrete sources is used to approximate the scattered field in the unbounded region outside the scattering body. A comparison of the numerical solution with an analytic solution is performed.     

Propagation of sound waves through a spatially homogeneous but smoothly time-dependent medium

The propagation of sound through a spatially homogeneous but non-stationary medium is investigated within the framework of fluid dynamics. For a non-vortical fluid, especially, a generalized wave equation is derived for the (scalar) potential of the fluid velocity distribution in dependence of the equilibrium mass density of the fluid and the sound wave velocity. A solution of this equation for a finite   transition period τ$\tau$ is determined in terms of the hypergeometric function for a phenomenologically realistic, sigmoidal change of the mass density and sound wave velocity. Using this solution, it is shown that the energy flux of the sound wave is not conserved but increases always   for the propagation through a non-stationary medium, independent of whether the equilibrium mass density is increased or decreased. It is found, moreover, that this amplification of the transmitted wave arises from an energy exchange with the medium and that its flux is equal to the (total) flux of the incident and the reflected wave. An interpretation of the reflected wave as a propagation of sound backward in time is given in close analogy to Feynman and Stueckelberg for the propagation of anti-particles. The reflection and transmission coefficients of sound propagating through a non-stationary medium is analyzed in more detail for hypersonic waves with transition periods τ$\tau$ between 15 and 200 ps as well as the transformation of infrasound waves in non-stationary oceans.     

Traffic noise and the hyperbolic plane

We consider the problem of sound propagation in a wind. We note that the rays, as in the absence of a wind, are given by Fermat’s principle and show how to map them to the trajectories of a charged particle moving in a magnetic field on a curved space. For the specific case of sound propagating in a stratified atmosphere with a small wind speed, we show that the corresponding particle moves in a constant magnetic field on the hyperbolic plane. In this way, we give a simple ‘straightedge and compass’ method to estimate the intensity of sound upwind and downwind. We construct Mach envelopes for moving sources. Finally, we relate the problem to that of finding null geodesics in a squashed anti-de Sitter spacetime and discuss the SO(3,1)×R symmetry of the problem from this point of view.     

On the statement of the problem of sound scattering by a cylindrical vortex

The well-known problem of sound wave scattering by a Rankine vortex is investigated. Although the problem has been studied for years, none of the solutions reported in the literature can be considered completely correct. It is demonstrated that the main difficulty consists in the absence of a mathematically well-posed statement of the problem for a plane wave (which is used in most of the approaches) because of the slow decrease in the mean flow velocity at infinity. This gives rise to multiple solutions, including those singular on a line behind the vortex, and each of them claims to be correct. It is shown that, in spite of the decrease in the mean flow velocity, the problem does not possess any remote region at infinity, tending to which it is possible to preset the plane wave condition for the incident field. Therefore, for the external field, the remote point source condition is proposed. This approach makes it possible to state a mathematically well-posed problem, to reveal the origin of the aforementioned ambiguity, and to compare previous approaches used for solving the problem under consideration.     

Gasdynamic design of a two-dimensional supersonic inlet with the increased flow rate factor

A two-dimensional inlet of external compression with the increased flow rate factor at high supersonic velocities is constructed by the method of gasdynamic design. Its feature is that a flow with the initial oblique shock wave and the subsequent centered isentropic compression wave is formed over the external compression ramp of the inlet. These waves interact with one another so that a resulting stronger oblique shock wave and a velocity discontinuity arise in front of the entrance to the inlet internal duct. An example of an inlet configuration with the design flow regime corresponding to the Mach number M_d = 7 is considered. The characteristics of this inlet were obtained in the range of the free-stream Mach numbers M = 4–7 with the use of a Navier—Stokes code for turbulent flow. They are compared with characteristics of an equivalent conventional shocked inlet. As computations have shown, the inlet with the isentropic compression wave has much higher values of flow rate factor φ at Mach numbers M < M_d. So, for example, at M = 4 the value φ ≈ 0.72 for it is by 33 % higher in comparison with φ ≈ 0.54 for the equivalent shocked inlet.     

On the scattering of sound by a vortex ring

An analytical treatment is given of the scattering of sound by a circular vortex ring translating at low Mach number in an unbounded stationary medium. The theoretical results complement the recent numerical investigation of this problem by Kambe and Mya Oo [1].     

Potential splitting approach to the three-body coulomb scattering problem

The potential splitting approach is extended to a three-body Coulomb scattering problem. The distorted incident wave is constructed and the driven Schrödinger equation is derived. The full angular momentum representation is used to reduce the dimensionality of the problem. The phase shifts for e⁺?H and e⁺?He⁺ collisions are calculated to illustrate the efficiency of the presented method.