On the characterization of breather and rogue wave solutions and modulation instability of a coupled generalized nonlinear Schrödinger equations

We construct Darboux transformation of a coupled generalized nonlinear Schrödinger (CGNLS) equations and obtain exact analytic expressions of breather and rogue wave solutions. We also formulate the conditions for isolating these solutions. We show that the rogue wave solution can be found only when the four wave mixing parameter becomes real. We also investigate the modulation instability of the steady state solution of CGNLS system and demonstrate that it can occur only when the four wave mixing parameter becomes real. Our results give an evidence for the connection between the occurrence of rogue wave solution and the modulation instability.     

Local and global instabilities of flow in a flexible-walled channel

We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability.     

用PIV测试涡旋波流场的速度和剪应力分布

采用PIV瞬态流场测试技术,对二维槽道中的涡旋波流场不同相位上的速度分布和应力分布进行了测试和计算,本文定量地描述了槽道中涡旋波的形成过程及发展规律。通过调整振荡流的振幅和活塞行程,分析了Re数和Sr数对涡旋波流动的影响,得出了槽道内剪应力的分布状况以及平均剪切应力的周期性变化规律。深入分析了涡旋波流场强化传递现象的本质。     

Mixing Layers in Sink Flow: Effect of Length of Flight on Mixing in a Channel Downstream

We have experimentally and analytically studied transport of a passive scalar in the wake of a thin flat plate located at the centerline of a planar contraction with flat walls. The constant Launder parameter in the contraction, K = 6.25 ×10^− 6, was twice the value required for a turbulent boundary layer to relaminarize. In addition to the mixing analysis inside the contraction, layer mixing is also investigated downstream, where the flow continues inside a constant cross-section channel. In order to generate the passive scalar, the airflow above the plate was heated and the temperature stratification in the wake was traced by measuring the temperature field using constant current anemometry. Using different plate lengths, we found that the degree of mixing, obtained at a given position in the straight channel, is a function of the distance from the plate trailing edge to the contraction outlet. For a plate which does not protrude into the straight channel, we demonstrate the existence of an optimal trailing edge-contraction outlet distance that results in the lowest possible degree of mixing at a given downstream position in the straight channel. This finding is also supported by a semi-empirical relationship based on our developed self-similar solution for mixing layers in planar contractions.     

Effects of geometric modulation and surface potential heterogeneity on electrokinetic flow and solute transport in a microchannel

A numerical investigation is performed on the electroosmotic flow (EOF) in a surface-modulated microchannel to induce enhanced solute mixing. The channel wall is modulated by placing surface-mounted obstacles of trigonometric shape along which the surface potential is considered to be different from the surface potential of the homogeneous part of the wall. The characteristics of the electrokinetic flow are governed by the Laplace equation for the distribution of external electric potential; the Poisson equation for the distribution of induced electric potential; the Nernst–Planck equations for the distribution of ions; and the Navier–Stokes equations for fluid flow simultaneously. These nonlinear coupled set of governing equations are solved numerically by a control volume method over the staggered system. The influence of the geometric modulation of the surface, surface potential heterogeneity and the bulk ionic concentration on the EOF is analyzed. Vortical flow develops near a surface modulation, and it becomes stronger when the surface potential of the modulated region is in opposite sign to the surface potential of the homogeneous part of the channel walls. Vortical flow also depends on the Debye length when the Debye length is in the order of the channel height. Pressure drop along the channel length is higher for a ribbed wall channel compared to the grooved wall case. The pressure drop decreases with the increase in the amplitude for a grooved channel, but increases for a ribbed channel. The mixing index is quantified through the standard deviation of the solute distribution. Our results show that mixing index is higher for the ribbed channel compared to the grooved channel with heterogeneous surface potential. The increase in potential heterogeneity in the modulated region also increases the mixing index in both grooved and ribbed channels. However, the mixing performance, which is the ratio of the mixing index to pressure drop, reduces with the rise in the surface potential heterogeneity.     

Guided waves in a moving layer of fluid between two elastic solids

We consider the motion of compressible fluid in a channel between two infinite elastic layers with the same properties (homogeneous and isotropic). We suppose that the layer of fluid moves with a constant velocity, respectively, to the other layers. Under certain conditions the moving layer can function as a waveguide. The aim is to determine necessary conditions for the existence of a running wave along the channel and a standing wave in a transversal direction when the channel functions as a waveguide.     

Phase dynamics of effective drag and lift components in vortex-induced vibration at low mass–damping

In this work, we investigate the dynamics of vortex-induced vibration of an elastically mounted cylinder with very low values of mass and damping. We use two methods to investigate this canonical problem: first we calculate the instantaneous phase between the cylinder motion and the fluid forcing; second we decompose the total hydrodynamic force into drag and lift components that act along and normal to, respectively, the instantaneous effective angle of attack. We focus on the phase dynamics in the large-amplitude–response range, consisting of the initial, upper and lower “branches” of response. The instantaneous phase between the transverse force and displacement shows repeated phase slips separating periods of constant, or continuous-drifting, phase in the second half of the upper branch. The phase between the lift component and displacement shows strong phase locking throughout the large-amplitude range – the average phase varies linearly with the primary frequency – however the modulation of this phase is largest in the second half of the upper branch. These observations suggest that the large-amplitude–response dynamics is driven by two distinct limit cycles – one that is stable over a very small range of reduced velocity at the beginning of the upper branch, and another that consists of the lower branch. The chaotic oscillation between them – the majority of the upper branch – occurs when neither limit cycle is stable. The transition between the upper and lower branches is marked by intermittent switching with epochs of time where different states exist at a constant reduced velocity. These different states are clearly apparent in the phase between the lift and displacement, illustrating the utility of the force decomposition employed. The decomposed force measurements also show that the drag component acts as a damping factor whereas the lift component provides the necessary fluid excitation for free vibration to be sustained.     

Supercritical Flow Over a Sill in an Open Channel

Experimental data on typical profiles of free surface and channel bottom pressure for a supercritical flow over a sill are reported. This flow is shown to have, along with the known critical depth, two other characteristic depths, one of which is at the channel exit to the atmosphere and the other determines conditions under which the disturbances propagate well upstream of the sill. The experimental data are compared with calculation results based on a mathematical model that incorporates turbulent mixing upon wave breaking.     

Promotion of Turbulent Thermal Mixing of Hot and Cold Airflows in T-junction

An experimental study has been conducted on the promotion and control of turbulent thermal mixing of hot and cold airflows in a T-junction with rectangular cross sections, which simulates the HVAC unit for automobile air-conditioning system. In order to promote the turbulent thermal mixing, small jets have been blown into the main channel at the upstream edge of the T-junction in the direction of 45° against the main flow. Turbulence intensity in the upper part of the thermal mixing layer can be increased with these jets, and consequently the turbulent mixing of hot and cold airflows is promoted effectively. Moreover, it has been found that the degree of thermal mixing can be controlled by changing the jet velocity.     

Identification of vortex motions in turbulent mixing of choked jets

A specially adapted schlieren system is used to generate fluctuating signals which respond strongly to large scale coherent components of a turbulent mixing jet flow and which have a relatively reduced response to random disturbances. The schlieren signals also provide a direct indication of the presence of vortex-like structures in the turbulent mixing layers by virtue of the phase relationship of the schlieren signals to the pressure field. This system gives a clear resolution of the fluctuating periodic effects associated with vortex structures in the flow from a choked convergent nozzle. It has thus been possible to determine that vortex-like eddies are associated with the feedback screech mechanism, and also generate periodic disturbances due to their passage through the diamond shaped wave structure in the flow. The regular disturbances in the flow move at 0.77 of the fully expanded flow velocity. Phase spectral observations demonstrate clearly the vortex like structure of coherent disturbances in the flow by virtue of the quadrature phase relation between the schlieren and microphone signals. Movement of the sensing microphone in the pressure field external to the flow shows disturbance propagation at the acoustic velocity, and also shows that disturbances at Strouhal numbers above 0.7 emanating from the inner mixing zone can be identified by an additional time delay to reach the microphone and only influence the microphone when it is located downstream of the flow sensing schlieren system due to confinement of pressure disturbances within Mach cones of the flow.     

Numerical modeling of gravity currents in inclined channels

The mixing and wave formation processes in gravity currents induced by the rupture of a vertical dam initially separating a heavy and a light liquid are studied for different channel inclination angles. The calculations are performed using the LES and RANS models. It is shown that when the heavy liquid moves down the channel slope, the longitudinal and transverse internal waves break and form turbulent mixing zones. When the heavy liquid ascends the slope, the wavy motion mode predominates.     

A PLIF and PIV study of liquid mixing enhanced by a lateral synthetic jet pair

In this paper, enhancement of mixing between two water streams of the same flow rate in a planar channel by means of a lateral synthetic jet pair is studied at a net flow Reynolds number of 83 using PLIF and PIV. The synthetic jet pair is operated 180° out-of-phase at a range of actuation frequencies and displacements, with the latter being characterized by the dimensionless stroke length. The extent of mixing is evaluated using PLIF data at a location further downstream in the mixing channel. It is found that at a fixed actuation frequency a higher dimensionless stroke length produces a better mixing, and as the actuation frequency increases a lower dimensionless stroke length is required to achieve a given mixing degree. At a sufficiently high frequency or dimensionless stroke length, a nearly homogenous mixing with a mixing degree greater than 0.9 can be obtained. A functional relationship between actuation frequency and dimensionless stroke length is also obtained by best fitting the experimental data, which can be used for selecting the synthetic jet operating conditions to ensure a good mixing. Furthermore, both PLIF and PIV results show that each synthetic jet actuation cycle produce two opposing vortex pairs, which play an important role in prompting mixing between the two fluid streams. The excellent mixing obtained at a high frequency or a high dimensionless stroke length is found to be largely caused by a strong interaction between these opposing vortex pairs.     

平面激波冲击并排液滴的三维数值模拟研究

采用三维守恒清晰界面数值方法, 研究平面激波冲击并排液滴的动力学过程. 研究的焦点在于激波接触液滴后的复杂波系结构生成, 以及并排液滴相互耦合作用诱导的单个液滴非对称界面演化. 首先, 分析并排液滴之间界面通道内的波系结构发展, 发现在冲击初期由于反射激波相交而形成新的反射激波以及马赫杆; 这些流动现象与液滴另外一侧 (非通道侧) 由激波反射所形成的弯曲波阵面截然不同, 而且所导致的液滴横向两侧流场差异是中后期冲击过程液滴两侧界面非对称演化的主要原因. 其次, 研究冲击中期时, 特别是入射激波已运动至液滴下游并远离并排液滴, 界面形态的演化过程和规律, 揭示通道下游出口处由于气流膨胀导致的界面闭合、以及随后气流阻塞导致的界面破碎等新的流动现象. 最后, 研究液滴间距对并排液滴相互作用的影响规律, 发现液滴间距大小与通道内压力峰值具有明显的关联关系. 研究表明, 更小的液滴间距不仅带来更大的压力峰值, 而且使得峰值出现的时间更早.      

控制存在时滞的线性系统主动控制的滑移模态方法

本文研究控制存在时滞的线性系统的滑模控制方法。在控制时滞量为采样周期的整数倍和非整数倍的两种情况下，通过采用零阶保持器，将包含时滞的连续系统转化为形式上不包含时滞的标准离散线性系统，然后分别进行切换函数和控制律的设计。所得出的切换面和控制律表达式中，除了含有当前的状态反馈外，还包含有前若干步控制项的线形组合。最后对某三自由度结构模型进行仿真计算，验证了所给控制方法的有效性。     

The Saddle-Node of Nearly Homogeneous Wave Trains in Reaction–Diffusion Systems

We study the saddle-node bifurcation of a spatially homogeneous oscillation in a reaction-diffusion system posed on the real line. Beyond the stability of the primary homogeneous oscillations created in the bifurcation, we investigate existence and stability of wave trains with large wavelength that accompany the homogeneous oscillation. We find two different scenarios of possible bifurcation diagrams which we refer to as elliptic and hyperbolic. In both cases, we find all bifurcating wave trains and determine their stability on the unbounded real line. We confirm that the accompanying wave trains undergo a saddle-node bifurcation parallel to the saddle-node of the homogeneous oscillation, and we also show that the wave trains necessarily undergo sideband instabilities prior to the saddle-node.     

Mass transfer enhancement by longitudinal vortices

From numerical and experimental investigations it has been recently established that convective heat transfer can be dramatically enhanced by the generation of longitudinal vortices in the flow. The phenomenological similarity between heat and mass transfer suggests that longitudinal vortices should increase also mass transfer. The mixing between two parallel streams of two components in a rectangular channel with and without a pair of rectangular winglets as vortex generators has been numerically investigated. The results show that one pair of vortex generators can increase the global mixing by more than 50 for laminar flow. This global mixing has been defined as the sum of the square of the differences of concentrations.     

A study of sloshing absorber geometry for structural control with SPH

A liquid sloshing absorber consists of a container, partially filled with liquid. The absorber is attached to the structure to be controlled, and relies on the structure's motion to excite the liquid. Consequently, a sloshing wave is produced at the liquid free-surface within the absorber, possessing energy dissipative qualities. The primary objective of this work is to numerically demonstrate the effect of a sloshing absorber's shape on its control performance. Smoothed Particle Hydrodynamics (SPH) is used to model fluid–structure interaction of the structure/sloshing absorber system in two dimensions. The structure to be controlled is a lightly damped single degree-of-freedom structure. The structure is subjected to a transient excitation and then allowed to respond dynamically, coming to rest either due to its own damping alone or with the added control of the sloshing absorber. It is identified that the control performance of the conventionally used rectangular container geometry can be improved by having inward-angled walls. This new arrangement is robust, and of significant advantage in situations when the external disturbance is of uncertain magnitude.     

A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level

Propagation of interfacial waves near the critical depth level in a two-layer fluid system is investigated. We first present a generalized modified Kadomtsev-Petviashvili (gmKP) equation for weakly nonlinear and dispersive interfacial waves propagating predominantly in the longitudinal direction of a slowly rotating channel with gradually varying topography and sidewalls. For certain type of non-rotating channels, we find two families of periodic-wave solutions, which include solitarywave solutions and a shock-like solution as limiting cases, to the variable-coefficient gmKP equation. We also show that in this situation the gmKP equation has only unidirectional N-soliton solutions and does not allow soliton resonance to occur. In a rotating uniform channel, our small-time asymptotic analysis and numerical study of the gmKP equation show that, depending on the relative importance of the cubic nonlinearity to quadratic nonlinearity, the wavefront of a Kelvin solitary wave may curve either forward or backward, trailed by a small train of Poincaré waves. When these two nonlinearities almost balance each other, the wavefront becomes almost straight-crested across the channel, and the trailing Poincaré waves diminish.