Convective instability of a system of horizontal layers of immiscible liquids with a deformable interface

The convective stability of equilibrium is considered for a system of two immiscible fluids which differ little in density. A generalized Boussinesq approximation is developed, making it possible to take the interface deformations properly into account. The stability of the equilibrium state of two fluids in a horizontal layer with a vertical temperature gradient is investigated. Several instability mechanisms are identified: long-wave and cellular monotonic disturbances and oscillatory disturbances. Increasing the deformability is shown to cause switching between instability mechanisms.Perm. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 32–39, March–April, 1996.     

Thermocapillary Instability of a Cylindrical Layer in the Presence of a Radial Temperature Gradient

Within a linear formulation, the thermocapillary instability of equilibrium of a cylindrical layer of heat-conducting viscous fluid in the presence of a radial temperature gradient is investigated with respect to arbitrary disturbances. It is shown that the Rayleigh instability mechanism results in the appearance of monotonous disturbances of a new type. For steady disturbances, the neutral curve is split into two separate segments, each corresponding to its own type of disturbances. For a deformable free boundary, new oscillating disturbances in the form of surface waves develop. It is found that, in the case of axial symmetry, the behavior of these disturbances completely coincides with the oscillating disturbance behavior in a plane layer.     

Self-sustained amplification of disturbances in pipe Poiseuille flow

The nonlinear development of disturbances in pipe Poiseuille flow is studied with a low-dimensional model. The basic system from which the model is derived governs disturbances closely related to the radial velocity and radial vorticity disturbances. The analysis is restricted to the interaction of the two first harmonics of streamwise elongated disturbances since they are the most transiently amplified ones in linear theory. In the resulting dynamical system a nonlinear feedback from the normal vorticity disturbance (which is transiently amplified according to linear theory) to the radial velocity disturbance is present. Above a threshold of the initial amplitude, the feedback leads to a self-sustained amplification of the disturbances continuing for all times.     

Effect of disturbances and sensorimotor deficits on the postural robustness of an ankle–hip model of balance on a balance board

This paper investigates the effect of postural disturbances and sensorimotor deficits on the robustness of the upright posture (UP) for a human body model balancing on a balance board (BB). The robustness is investigated by computing the gradient field along the basin of attraction (BoA) of an asymptotically stable equilibrium point. The human model is modeled as a double-inverted pendulum (hip and ankle joints). The human-BB system is assumed actuated by torques at the hip, BB hinge, and ankle joints. Postural disturbances induce an initial joint angle velocity either at the ankle or at the hip joint. Moreover, either proprioceptive or visual and vestibular deficits are considered in the human-BB model. The nonlinear dynamic equation of motion of the human-BB system is numerically solved to obtain the BB, ankle, and hip joint angle position and velocity profiles. The BoA of the human-BB UP equilibrium is built as the set of initial conditions whose resulting time-series profiles converge to the equilibrium. It was shown that UP is more robust to disturbances that induce hip joint initial angle velocity. That is probably due to the fact that disturbances that induce ankle joint initial velocity affect the whole body, while disturbances that induce hip joint angle initial velocity only affect the trunk. Whenever visual and vestibular deficits are considered, the UP is more robust if proprioceptive gain and BB stiffness are small. Contrarily, whenever proprioceptive deficits are considered, the UP is more robust if visual and vestibular gain and BB stiffness are large. The method proposed here (the BoA and the gradients) can be used to systemically provide understanding about the robustness of the human-BB UP to external disturbances, which may help to identify people with a higher risk of fall.     

UDE-based decoupled full-order sliding mode control for a class of uncertain nonlinear MIMO systems

In this paper, a generalized control scheme for the class of nonlinear multiple-input multiple-output (MIMO) uncertain system with cross-coupling and nonlinearity in their input channels under the influence of external disturbances is presented. This is accomplished using full-order model following sliding mode control based on uncertainty and disturbance estimator (UDE) technique. The fourth-order uncertain nonlinear MIMO system is separated into multiple single-input single-output double integrator subsystems by considering the effect of input coupling and nonlinearity as a disturbance. The UDE is designed to estimate the plant uncertainties as well as external disturbances without the knowledge of the bounds on the uncertainties. The proposed method decouples the system and overcomes the problem of high initial control which ultimately eliminates the reaching phase and the chattering phenomenon which is generally occurred in sliding mode control. The effectiveness of the proposed control scheme is demonstrated through numerical simulation of two-link manipulator.     

Development of wedge-shaped disturbances originating from a point source in a three-dimensional boundary layer

A recent study predicted possibility of existence of a new instability due to the curvature of external streamlines in three-dimensional boundary layers, besides the familiar cross-flow instability, but no reliable evidence of this phenomenon has yet been obtained in experiments. In expectation of dispersive development of the two instabilities, the present study deals with small disturbances induced by continuous forcing from a point source in the boundary layer along a yawed circular cylinder, and attempts to describe their spatial development into wedge-shaped distribution with a linear stability theory, which is applicable to both of the above instabilities. Unlike plane-wave disturbances, the point-source disturbances have an important peculiarity that their propagation is governed by a complex group velocity, and a new method based on the complex property of the group velocity is presented to predict the paths of propagation along which growth rates of disturbances are integrated. Results of this stability calculation clarify important differences in development between the cross-flow disturbances and the streamline-curvature disturbances. These differences will make it possible to observe the new mode of disturbances separately from the other in experiments.     

Wave generation on an interface by vortex disturbances in a shear flow

A linear problem of oscillations of an interface in a two-layer system, in which the upper layer is at rest and the lower layer has a constant velocity shear, is considered. The dynamic perturbations in the lower layer are represented as the sum of vortex and wave disturbances (disturbances with zero vorticity). It is shown that in the shear flow the evolution of the vortex disturbances with a nonsmooth or a singular initial vorticity distribution can result in the resonant excitation of waves on the interface. The occurrence of the resonance corresponds to the coincidence of the oscillation frequencies of the perturbations of both classes. In the absence of hydrodynamic instability of the shear flow, the resonant excitation can be one of the main mechanisms of wave generation in two-layer systems.     

Dynamics of nonlinear three-dimensionalwaves on the interface between two fluids in a channel with low-sloping bottom and top

A combined approach is proposed to describe the transformation of three-dimensional disturbances of the interface between two incompressible immiscible fluids of different densities contained in a channel with fixed rigid top and bottom. It is assumed that the wavelengths are moderately large, the amplitudes are small but finite, the top and the bottom can be gently sloping, and capillary effects are small. The system of equations derived is applicable for modeling disturbances simultaneously scattering in arbitrary horizontal directions. Some typical wave problems are numerically solved and the effect of governing parameters is shown.     

Stability analysis of unbounded uniform shear flows of dense, slightly inelastic spheres based on a frictional-kinetic theory

Asymptotic and transient stability analyses of unbounded uniform shear flows of dense, slightly inelastic, spherical particles were carried out using a frictional-kinetic theory. This model proposed for describing dense flows is based on a critical state plasticity theory and a simplified kinetic theory. In this model, the bulk and shear viscosities, the “thermal” conductivity, and the energy dissipation rate are proportional to a “mean pressure” which is composed of a quasistatic-frictional-contribution pressure considered for slow, plasticity deformations and a granular-kinetic-theorycollisional-contribution pressure. We studied two-dimensional stability analyses of layering disturbances (i.e., the perturbations whose wave number vectors are aligned only in the gradient) as well as nonlayering disturbances (the wave number vectors have nonzero streamwise components). Although this model has a simpler framework, it predicted similar results to those obtained using a more elaborate frictional-kinetic model. For instance, nonlayering disturbances are asymptotically stable at large time; the maximum transient growth of disturbances increases as the solids fraction or the friction coefficient is increased; and transient growths of disturbances can be significant due to the non-normality of the system. However, the prediction of the asymptotic stability of layering disturbances may be questionable because the collisional-contribution terms of the present model were over-simplified.     

Stability of equilibrium of a double-layer system with a deformable interface and a prescribed heat flux on the external boundaries

The stability of mechanical equilibrium of a system of two horizontal immiscible-liquid layers with similar densities is studied. The problem is solved for a prescribed heat flux on the external boundaries. Within the framework of a generalized Boussinesq approximation, which takes the interface deformation correctly into account, the onset of convection caused by heating the system from above or below is considered. Two long-wave instability modes attributable to the presence of the deformable interface and the given heat flux on the external boundaries are detected. The system response to monotonic and oscillatory disturbances with finite wavelengths is investigated. A complete stability map is constructed.     

Hydrodynamic and thermal characteristics of a two-phase transpiration cooling system

This paper generalizes static characteristics of heat engineering processes which tend to instability. Hydrodynamic and thermal characteristics of a two-phase transpiration cooling system are plotted based on the solution of a nonlinear closed system of differential equations describing the hydrodynamics and heat transfer of a filtering coolant with unknown position of the surface of its equilibrium phase transformation. The study of aperiodic stability of two-phase transpiration cooling with different static characteristics yields the same results, while the use of each characteristic separately makes it possible to detect permissible disturbances of an appropriate characteristic parameter. With the use of a set of static characteristics, permissible disturbances of all of the characteristic parameters can be found and the peculiarities of stable two-phase transpiration cooling can be revealed.     

Three-wave interactions of disturbances in a hypersonic boundary layer on a porous surface

Interactions of disturbances in a hypersonic boundary layer on a porous surface are considered within the framework of the weakly nonlinear stability theory. Acoustic and vortex waves in resonant three-wave systems are found to interact in the weak redistribution mode, which leads to weak decay of the acoustic component and weak amplification of the vortex component. Three-dimensional vortex waves are demonstrated to interact more intensively than two-dimensional waves. The feature responsible for attenuation of nonlinearity is the presence of a porous coating on the surface, which absorbs acoustic disturbances and amplifies vortex disturbances at high Mach numbers. Vanishing of the pumping wave, which corresponds to a plane acoustic wave on a solid surface, is found to assist in increasing the length of the regions of linear growth of disturbances and the laminar flow regime. In this case, the low-frequency spectrum of vortex modes can be filled owing to nonlinear processes that occur in vortex triplets.     

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.     

Tracking differentiator and extended state observer-based nonsingular fast terminal sliding mode attitude control for a quadrotor

Nonlinear Dynamics - The attitude control problem is addressed for a quadrotor system subject to the modeling uncertainties and unknown disturbances. A novel attitude control scheme is proposed...     

Propagation of wave packets in the boundary layer on a curved surface

The development of wave packets excited in a boundary layer by means of a local deformation of the surface in the longitudinal-transverse interaction regime is considered. A solution of the linearized system of equations of interaction theory is constructed using a Laplace transformation with respect to time and a Fourier transformation with respect to the space variables. Two problems are separately examined. In the first, the disturbances are induced by a surface deformation sinusoidal in the transverse direction. It is shown that the center of the wave packet with the greatest oscillation amplitude moves in a direction opposite to that of the flow in the boundary layer. At the same time the wave packet expands, so that in the course of time any fixed point will enter the region of growing oscillations. In the second problem the source of the disturbances is isolated. In this case the wave packet acquires a horseshoe shape. Expanding, it carries the disturbances away from the source in all directions, including upstream relative to the flow in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 59–68, March–April, 1990.     

The design of nonlinear disturbance attenuation controller for TCSC robust model of power system

The control problem of the single machine infinite bus system with TCSC is dealt with. Based on the maximization of the external disturbances on the system model, an adaptive nonlinear controller for large disturbance attenuation and a parameter updating law are designed by using the backstepping method. The parameter uncertainty of the transmission line is considered, as well as the influences of large external disturbances to the system output are mainly discussed. The nonlinear controller does not have the sensitivity to the influences of external disturbances, but also has strong robustness for system parameters variation. The simulation results show that the control effect of the large disturbance attenuation controller more advantages by comparing with the control performance of conventional nonlinear robust controller.     

Development of Natural Disturbances in a Hypersonic Boundary Layer on a Sharp Cone

Experimental data on the location of the laminar—turbulent transition and development of natural disturbances in a laminar hypersonic boundary layer on a sharp thermally insulated cone with a half–angle of 7° are presented. The existence of the second mode of disturbances is confirmed. It is shown that the transition is determined by the first mode of disturbances. The experimental data are in good agreement with theoretical calculations.     

Disturbance spectrum of a plane fluid-fluid interface in the field of tangential high-frequency vibrations underweightlessness

The spectra of time-dependent disturbances of the equilibrium state and the stability of a plane interface between two fluid layers having different but comparable densities and being in the field of tangential high-frequency vibrations under weightlessness are investigated. Plane, spiral, and three-dimensional disturbances are considered. The cases of the same and different layer thicknesses are analyzed (in the latter case one of the layers is thicker than the other by a factor of ten). It is established that it is monotonic plane disturbances that are most hazardous. It is found that at high values of a vibration parameter growing spiral oscillatory disturbances (traveling waves) appear. With intensification of the vibrations the oscillatory disturbances vanish from the spectrum. In the case of the layers of different thicknesses it is established that the wavelength of the most hazardous disturbances is of the order of the thinner layer thickness. The experimentally observed generation of alternating strata in two-layer systems in the high-frequency vibration filed under weightlessness is attributed to the growth of disturbances having the greatest growth rate. The results obtained are quantitatively compared with the experimental data.     

Associated disturbances around a vortex ring in a stratified fluid

A motion of a vortex ring in a stratified fluid is accompanied by associated disturbances which, in the schlieren visualization in the field of a horizontal density gradient, have the shape of a symmetric four-petal configuration. The criterion of the existence of the disturbances is the Froude number Fr based on the motion velocity and the vertical vortex size. On the range Fr > 1, the disturbances are stable with respect to the variation of themotion regime and the distortion of the vortex shape. For Fr < 1 the disturbances disappear. Computer processing of the schlieren photographs showed that the experimental spatial dependences of the disturbance amplitude are close to the functions describing the distribution of the vertical velocity component in the inviscid flow past a sphere.     

Numerical study of self-induced wing rock of a low-aspect-ratio delta wing in the presence of external disturbances

Self-induced wing rock of a delta wing, in particular, in the presence of external disturbances are studied by means of numerical simulations of a separated flow of an ideal incompressible fluid around a delta wing. The results obtained are compared with experimental data. The vortex nature and the mechanism of self-induced oscillations are studied. Regions of synchronization of the aerodynamic self-oscillatory system in the presence of external disturbances are identified. Methods of suppression of self-induced wing rock are proposed.