Three-wave interactions between disturbances in the hypersonic boundary layer on impermeable and porous surfaces

The interaction between disturbances in the hypersonic boundary layer on impermeable and porous surfaces is considered within the framework of weakly-nonlinear stability theory. It is established that on the impermeable surface nonlinear interactions between different waves (acoustic and vortex) occur in the parametric resonance regime. The role of pumping wave is played by a plane acoustic wave. The nonlinear interactions take place over a wide frequency range and can lead to the packet growth of Tollmien-Schlichting waves. On the porous surface the analogous interactions are fairly weak and result in a slight decay of the acoustic mode and a slight amplification of the vortex mode. This leads to the dragging out of the laminar flow regime and the regions of linear disturbance growth. In this situation the low-frequency spectrum of the vortex modes may be filled on account of the nonlinear processes occurring in the three-wave systems between the vortex components.     

蜿蜒边界下平面流的线性流动稳定性

为便于数值分析,蜿蜒河流水动力和演变模型中一般隐性假设二次时均流-二次涡的关系与明渠流时均流-明渠湍流的关系相同,但由于高雷诺数下的DNS算力限制和实验尺度限制,这种隐含假设是否成立目前尚无相关湍流研究来支撑.文章试图通过分析明渠湍流和二次湍流发展初期的研究,侧面揭示其湍流结构的异同.通过对曲线正交坐标系下的平面二维NS方程使用双参数摄动的方法,建立了一种求解蜿蜒边界弱非线性层流的摄动解法,并推导得出一个适用于蜿蜒边界的EOS方程以及其特征值问题的解法.蜿蜒边界下弱非线性层流解为一系列蜿蜒谐波分量的叠加,其中线性部分使得两壁产生流速差,非线性部分随着雷诺数增大呈指数增长.水流的扰动增长率特征谱的第一模态与直道流相似,由3条曲线、4个波段合成,但其长波段和短波段的扰动流场与直道流不同,所有短波段的扰动流速近似于KH涡.蜿蜒边界对内部水流扰动有一定的选择性.偏角幅值越大扰动增长越快;蜿蜒波数的影响则为先增后减,有一个使扰动增长最快的蜿蜒波数.扰动流场由一个典型的TS波和一对波包形式的二次涡叠加而成,波包只有纵向流速分量,包络线由蜿蜒波数控制,波包内是与直道扰动波参数相同的TS波.     

Packets of wave perturbations in a Blasius flow

Asymptotic methods are used in order to study the development of quasiharmonic perturbations in the linear and slightly nonlinear problems of the stability of locally parallel Blasius flow. The connection between complex frequencies and wave numbers is analyzed in dependence on the initial spectrum and the local Reynolds numbers R, the law of propagation of the packets is determined, the equation of the amplitudes of their envelopes is constructed and solved, and an explanation is given of the conditions for the activation of regimes of the regular and irregular types. An interpretation is given of the observed properties of the initial stage in the evolution of the pulse in the boundary layer. It is concluded that the model has a limited ability to explain the mechanisms of stochastization.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 32–38, January–February, 1988.     

Frequency-dependent acoustic energy focusing in hexagonal ceramic micro-scaffolds

Acoustic properties of an additive-manufactured SiC scaffold with hexagonal symmetry fabricated by the robocasting method are studied both numerically and experimentally. The numerical analysis is based on the finite element method (FEM) using Bloch boundary conditions. The calculations show both angular and frequency dispersion of the acoustic waves with wavelengths comparable to the spacing between the rods, i.e., on a millimeter scale, indicating interesting acoustic properties in the MHz range. The dispersion character leads to focusing of the energy propagation into the directions of the rods of the hexagonal structure. This is illustrated by modal-based calculations of the propagation of longitudinal and out-of-plane shear wave packets with a dominant wavelength. The experimental analysis consists of two steps, the measurement of the resonant spectrum and shear wave propagation character. The measured resonant spectrum is in good agreement with the one calculated using numerically obtained low-frequency properties of the structure, also showing the quality of the overall manufactured structure. The time-domain measurement shows significant changes in the energy propagation between low and high frequencies, as predicted by FEM calculations.     

Parametric interaction of acoustic waves in micro-inhomogeneous media with hysteretic nonlinearity and relaxation

A theoretical investigation of parametric processes that arise as a result of the interaction of powerful and weak longitudinal acoustic waves in micro-inhomogeneous media with hysteretic nonlinearity and relaxation was carried out. The case of degenerate interaction between a powerful high-frequency wave and a weak low-frequency one was considered. The nonlinear damping coefficient and the carrier frequency phase delay of the weak wave propagating under the action of the powerful wave were determined.     

Evolution of laminar flow disturbances behind a step on a surface generated by its localized vibrations

The formation and development of hydrodynamic disturbances generated by low-frequency vibrations of a local region on a flat plate behind a rectangular step in separated flow is investigated in a wind tunnel. The results are obtained at a small subsonic flow velocity using the hot-wire anemometry. It is established that the wall vibrations induce separation zone disturbances representing streaky structures accompanied by wave oscillation packets. Laminar boundary layer separation favors the wave packet growth followed by wall flow turbulization.     

The influence of topography on the nonlinear interaction of Rossby waves in the barotropic atmosphere

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Three-Wave Interactions of Disturbances in a Supersonic Boundary Layer

Within the framework of the weakly nonlinear stability theory, group interaction of disturbances in a supersonic boundary layer is considered. The disturbances are represented by two spatial packets of traveling instability waves (wave trains) with multiple frequencies. The possibility of energy redistribution in such wave systems in the case of three-wave resonant interactions of packet constituents is considered. The model is used to test the dynamics of unstable waves arising due to introduction of controlled high-intensity disturbances into a supersonic boundary layer. It is found that this mechanism is not the main one for the features of streamwise dynamics of such nonlinear waves being observed.     

Application of the Discrete Fourier Transform to Study the Dynamics of Wave Packets

A method for solving equations that describe the dynamics of wave packets of the Tollmien–Schlichting waves in the boundary layer is proposed. The method of splitting the initial problem into the linear and nonlinear parts at each time step is used. The linear part is resolved by using an equation for spectral components of the wave packet with a subsequent Fourier transform from the space of wavenumbers to the physical space. A system of ordinary differential equations is solved in the physical space. The Fourier transform is performed by means of the library procedure of the fast Fourier transform. As examples, the problems solved were the linear dynamics of the wave packet concentrated in the vicinity of the instability region (i.e., a set of wave vectors in the space of wavenumbers for which the imaginary part of the eigenfrequency of the Tollmien–Schlichting waves is positive) and the nonlinear dynamics of the wave packet overlapping the instability region.     

Method of factorization in unsteady problems of gravity-elastic wave excitation in a fluid with a partially free boundary

An approach to the solution of unsteady problems with mixed boundary conditions for a layer of heavy fluid is developed. The plane problem of wave excitation by displacements given in a certain region of the lower boundary of the layer when the upper boundary is partially covered by an elastic plate is examined by way of illustration. As distinct from [1, 2], the proposed approach makes it possible to construct a solution in the form of a sum of harmonics and to carry out an analytic investigation into the nature of the propagation and stabilization of the wave fields. The space-time regions of the forming and formed wave packets are identified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 100–106, November–December, 1985.     

激波/湍流边界层干扰压力脉动特性数值研究

激波/湍流边界层干扰问题广泛存在于高速飞行器内外流动中, 激波干扰会导致局部流场出现强压力脉动, 严重影响飞行器气动性能和飞行安全. 为了考察干扰区内脉动压力的统计特性, 对来流马赫数2.25, 激波角33.2°的入射激波与平板湍流边界层相互作用问题进行了直接数值模拟研究. 在对计算结果进行细致验证的基础上, 分析比较了干扰区外层和物面脉动压力的典型统计特征, 如脉动强度、功率谱密度、两点相关和时空关联特性等, 着重探讨了两者的差异及其原因. 研究发现, 激波干扰对外层和物面压力脉动的影响差异显著. 分离区内脉动以低频特征为主, 随后再附区外层压力脉动的峰值频率往高频区偏移, 而物面压力脉动的低频能量仍相对较高. 两点相关结果表明, 外层和物面脉动压力的展向关联性均明显强于其流向, 前者积分尺度过激波急剧增长随后缓慢衰减, 而后者积分尺度整体上呈现逐步增大趋势. 此外, 时空关联分析结果指出, 脉动压力关联系数等值线仍符合经典的椭圆形分布, 干扰区下游压力脉动对流速度将减小, 外层对流速度仍明显高于物面.      

冲击波测试系统低频特性与补偿方法研究

为提高冲击波超压峰值的测量精度,多数学者把重点集中在系统高频特性研究上,以拓宽带宽的方式提高峰值测试的准确性。冲击波另外两个主要参数正压作用时间、比冲量却和测试系统的低频特性息息相关。针对实爆中出现的不同传感器正压作用时间差异较大的问题,对冲击波信号进行了边际谱分析,获得了信号的低频特性。建立了一阶参数模型来表征低频特性,通过激波管试验数据获取了7种系统的低频模型参数。采用零极点配置法设计了低频补偿模型。结果表明:冲击波测试系统低频特性严重影响冲击波信号正压作用时间测试准确性,基于低频特性补偿的数据处理方法可以有效的提高冲击波信号正压作用时间、比冲量地测试精度。     

Intense bending waves in a bar

Summary In this paper, the work presented in [1] is extended to study higher-order approximations of nonlinear effects in a bar. It has been found that long bending waves, being the low-frequency modes involved in resonant triads, are stable against small perturbations. Consequently, a bending wave with group velocity which is less than that of longitudinal waves should behave as a linear quasi-harmonic wavetrain. On the other hand, one may expect self-modulation instability of intense bending wavetrains during the long-time evolution. This paper overcomes such a contradiction. To describe the nonlinear dynamics in detail, one should allow for higher-order approximation effects in the model. Such effects are associated with the diffusion of linear wave packets due to different group velocities, and amplitude dispersion caused by nonlinearity. Within the second-order approximation analysis, an amplitude modulation is indeed experienced for intense bending waves. As a result, envelope solitons can be formed from unstable bending wavetrains. The group matching of long longitudinal and short bending waves, being a particular case of the self-modulation, is of special interest as a limit case of the triple-wave resonant interactions. It demonstrates the relation between the first- and the second-order approximation effects. Accepted for publication 20 July 1996     

Self-demodulation acoustic signatures for nonlinear propagation in glass beadsSignatures acoustiques de l'auto-démodulation basse fréquence lors de la propagation non linéaire dans des billes de verres

The present Note describes some experimental work related to the nonlinear propagation of acoustic waves in granular media such as unconsolidated glass beads. The studied nonlinear effect is a self-demodulation process performed with the operation of the so-called parametric transmitting antenna. The pump (or carrier) wave is generated by a high power ultrasonic broad-band transducer (100 kHz central frequency) which is LF (low frequency, i.e., a few kHz) amplitude modulated. As the attenuation of acoustic waves increases with frequency, only the LF demodulated wave can be transmitted. A parametric study is performed where the HF central frequency is monitored between 60 and 300 kHz. The LF demodulation profile versus the HF frequency is modified, its shape being temporally derived almost twice. A numerical analysis of the order of temporal derivation is done in the Fourier domain, its value varying from 1.25 to 2.7. Qualitative agreement with current theoretical models is described, and an advanced theoretical analysis by the same authors [Phys. Rev. E 66 (2002) 041303], taking into account absorption, nonlinearity, dispersion and scattering, is briefly discussed. To cite this article: V. Tournat et al., C. R. Mecanique 331 (2003).     

Leading-edge receptivity of boundary layer to three-dimensional free-stream turbulence

The laminar-turbulent transition has always been a hot topic of fluid mechanics. Receptivity is the initial stage and plays a crucial role in the entire transition process. The previous studies of receptivity focus on external disturbances such as sound waves and vortices in the free stream, whereas those on the leading-edge receptivity to the three-dimensional free-stream turbulence(FST), which is more general in the nature,are rarely reported. In consideration of this, this work is devoted to investigating the receptivity process of three-dimensional Tollmien-Schlichting(T-S) wave packets excited by the three-dimensional FST in a flat-plate boundary layer numerically. The relations between the leading-edge receptivity and the turbulence intensity are established, and the influence of the FST directions on the propagation directions and group velocities of the excited T-S wave packets is studied. Moreover, the leading-edge receptivity to the anisotropic FST is also studied. This parametric investigation can contribute to the prediction of laminar-turbulent transition.     

On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves

The evolution of a three-dimensional packet of weakly non-linear internal gravity waves propagating obliquely at an arbitrary angle to the vertical line is considered. Two coupled non-linear equations connecting variations of a packet amplitude and induced flows are derived. three-dimensionality of the packet having been found responsible for the non-linearity of the system. Explicit formulae for the induced flow vertical component and the mean density field variation caused by packet propagation have been obtained. The plane wave is shown to be unstable at any arbitrary slope of the wave vector. The non-linear equation describing the evolution of the two-dimensional packet is derived in the subsequent order of the asymptotic scheme.It has been found possible for the packet to collapse. The collapse of internal waves packets may be one of the possible mechanisms of “blini”-shaped regions of mixed waters formation in the ocean.     

A class of coupled nonlinear Schr(o)dinger equations: Painlevé property,exact solutions,and application to atmospheric gravity waves

The Painlev(e) integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.     

Interaction of Coherent States for Hartree Equations

We consider the Hartree equation with a smooth kernel and an external potential, in the semiclassical regime. We analyze the propagation of two initial wave packets and show different possible effects of the interaction, according to the size of the nonlinearity in terms of the semiclassical parameter. We show three different sorts of nonlinear phenomena. In each case, the structure of the wave as a sum of two coherent states is preserved. However, the envelope and the center (in phase space) of these two wave packets are affected by nonlinear interferences, which are described precisely.