On the stability of generalized hill''s equation with three independent parameters

This paper presents a stability criterion of a generalized Hill's equation with three independent parameters, examined by means of an asymptotic method. The theory has been applied to a special form of Hill's equation and the region of stability, has been graphically illustrated. The separatrix between bounded and unbounded solutions yields in parametric space the contour of a ‘stabilitatskörper’ in which the solution is always stable.     

压气机流动稳定性自适应控制方法研究进展

压气机流动稳定性自适应控制是未来智能航空发动机的一项关键技术. 基础研究需要回答3个关切: 如何描述系统的稳定性？如何改变系统的稳定性？如何监测系统的稳定性？为此, 本团队在压气机流动稳定性通用理论、壁面阻抗边界扩稳方法和在线实时失速预警技术等3个方面开展了系统深入的研究工作. (1)所发展的叶轮机流动稳定性通用理论既能包含流动非均匀性又能考虑叶片几何, 计算高效, 预测精度高, 为压气机气动/稳定性一体化设计提供了可靠的评估工具. (2)所发展的基于壁面阻抗边界调控策略的SPS (stall precursor-suppressed)机匣处理和泡沫金属机匣处理在扩稳、降噪和保持系统气动性能方面取得实质性进展, 采用等价分布源方法建立了包含机匣处理影响的压气机失速起始预测模型, 对SPS机匣处理和泡沫金属机匣处理关键结构参数进行敏感性分析, 使其具有明确的理论设计准则. 实验结果证实, SPS机匣处理通过抑制失速先兆波的非线性演化达到扩稳的目的, 在扩稳的同时可以保持压气机的压比和效率特性; 泡沫金属机匣处理可以实现扩稳和降噪的双重效果, 也具有良好的工程应用前景. (3)所发展的基于气动声学原理的实时失速预警方法将压气机失速预警时间提高到秒量级以上, 能够在线监测系统稳定性. 综合上述理论预测方法、扩稳技术和实时失速预警技术, 发展了闭环反馈自适应控制方法, 为未来智能航空发动机提供了一种自适应扩稳控制技术.      

Vortex body interaction in a jet—circular cylinder sound generating system

An experimental investigation was conducted on relations between a ‘sound’ ^* and vortices generated by a plane Jet impinging on a circular cylinder. The ‘sound’ was a pseudo sound or near-field pressure fluctuations because its measurement was made at positions not sufficiently far from the Jet and the cylinder. The vortices consisted of vortices formed in the impinging jet and vortices shed from the cylinder. A thin splitter plate was attached to the downstream side of the cylinder to modify the shed vortices. The nature of the shed vortices was shown to have a significant influence on the intensity and frequency of the ‘sound’. The intensity of the ‘sound’ was found to be related to the streamwise extent of distributed vorticity within the vortices; that is, the more compact the vortices, the stronger was the ‘sound’.     

Forced oscillations of a piecewise-linear non-linear dynamic system with several degrees of freedom

Exact periodic solutions are derived for a dynamic system with several degrees of freedom consisting of a series of ‘Reid springs’ with piecewise-linear, non-linear characteristis; however, the solutions are restricted to a class of harmonic excitation in the ‘modal form’ described subsequently in the paper. Conditions are derived for the asymptotic stability of the periodic solution and an example has been worked out in detail on the response of a dynamic system with two degrees of freedom.     

Asymptotic soliton train solutions of Kaup–Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h₀(x) and u₀(x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u₀(x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.     

Perturbed vortical layers and shear sheltering

New theoretical results and physical interpretations are presented concerning the interactions between different types of velocity fields that are separated by thin interfacial layers, where there are dynamically significant variations of vorticity across the layers and, in some cases within them. It is shown how, in different types of complex engineering and environmental flow, the strengths of these interactions vary from the weakest kind of superposition to those where they determine the flow structure, for example by mutual exclusion of velocity fields from the other region across the interface, or by local resonance near the interface. We focus here on the excluding kinds of interactions between, on the one hand, elongated and compact regions containing vortical flows and large variations in velocity, and on the other hand various kinds of weak perturbation in the surrounding external flow region: rotational, irrotational; time-varying, steady; large, small; coplanar, non-coplanar; non-diffusive, diffusive. It is shown how all these kinds of external disturbances can be wholly, or partially, ‘blocked’ at the interface with the vortical region, so that beyond a certain sheltering distance into the interior of this region the fluctuations can be very small. For the special case of quasi-parallel co-planar external straining motions outside non-directional shear flows, weak sheltering occurs if the mean velocity of the shear flow increases – otherwise the perturbations are amplified. For non-parallel flows, the sheltering effect can be greater when the vorticity is distributed in thin vortex sheets. The mechanism whereby the vortical flow induces ‘blocking’ and ‘shear-sheltering’ effects can be quantitatively explained in terms of the small adjustments of the vorticity in the vortical layers, and in some cases by the change in impulse of these layers. If the vorticity in the outer part of the vortical region is weak, it can be ‘stripped away’ by the external disturbances until the remaining vorticity is strong enough to ‘block’ the disturbances and shelter the inner flow of the vortical region. The mechanisms presented here appear to explain on the one hand some aspects of the observed robustness of vortical structures and jet or plume like shear flows in turbulent and geophysical flows, and on the other hand the levels of external perturbation needed to erode or breakdown turbulent shear flows.     

Some new characteristics of the confined flow over circular cylinders at low reynolds numbers

This study summarises some new characteristics of the fluid flow over a confined circular cylinder at low Reynolds numbers. Results from both two- and three-dimensional direct numerical simulations are presented at blockage ratio between 0.1 and 0.9 and Reynolds number between 120 and 500. Floquet stability analysis of selected cases will also be presented. From the two-dimensional simulations, it is found that the fluctuating lift forces decreases with blockage ratio and becomes zero (where the flow is steady) at blockage ratio of approximately 0.7–0.8. Upon further increasing the blockage ratio to 0.9, the simulations show a dramatic increase in the fluctuating lift forces, nearly an order of magnitude greater than previously reported for an unconfined cylinder flow. It is also found that for blockage ratio of 0.5, there is a long term two-dimensional instability that becomes more prominent with increasing Reynolds number. This instability has a time scale of approximately 105 time units ($D/U_{\mathit{max}}$) at Reynolds number of 500. In addition, the transition between two- and three-dimensional flow at blockage ratios up to 0.5 is investigated. It is shown that the transition Reynolds number decreases with increasing blockage ratio. At high blockage ratio of 0.5, as we increase the Reynolds number, the transition to three-dimensional flow is shown to go from unsteady two-dimensional to steady three-dimensional before transitioning to unsteady three-dimensional flow.     

Circularly polarized plane waves in a deformed Hadamard material

Small-amplitude inhomogeneous plane waves propagating in any direction in a homogeneously deformed Hadamard material are considered. Conditions for circular polarization are established. The analysis relies on the use of complex vectors (or bivectors) to describe the slowness and the polarization of the waves.

Generally, homogeneous circularly polarized plane waves may propagate in only two directions, the directions of the acoustic axes, in a homogeneously deformed Hadamard material. For inhomogeneous circularly polarized plane waves, the number of possibilities is far greater. They include an infinity of ‘transverse waves’, as well as ‘longitudinal waves’, and the superposition of transverse waves and longitudinal waves, where ‘transverse’ and ‘longitudinal’ are used in the bivector sense.

Each and every possibility of circular polarization is examined in turn, and explicit examples of solutions are given in every case.     

Comparison of the statistics of two fracture modes

T strengths of an aggregate material are calculated for the following two modes of fracture: (i) by independent fracture of individual volume elements separated by a crack propagation barrier, (ii) by propagation of a crack from the weakest volume element in the absence of a propagation barrier. The first case is analysed by means of a model which consists of an assembly of layers in series, each layer representing a bundle of volume elements uniformly loaded in parallel. The strength distribution of the stacked layers is obtained by an application of the statistical theory of bundles of filaments after Daniels and others, and ‘weakest link’ statistics are used to yield the strength distribution of the cylindrical bodies. For comparison, the ‘weakest link’ statistics of Epstein and others are given for the second case which corresponds to the usual cleavage fracture model of brittle materials. The expected volume and shape effects of the two modes are compared.     

An actuator-disc model for the prediction of abrupt stall in an axial compressor rotor

The representation of loss in a cascade by the appearance of blockage has been extended to deal with blade rows by the use of a source distribution to represent this blockage, and in the case of the actuator disc approximation, the presence of sources is confined to an axi-symmetric dìstribution over the actuator disc. It is found that if a typical dependence of loss (and consequently diffusion ratio) upon incidence for each section of an axial compressor rotor is represented in this manner, the influence of blockage on the axial velocity distribution may be found using the potential equation combined with the usual actuator disc approximation. Study of the behaviour of the controlling ordinary differential equation for the axial velocity ahead of the disc reveals that as the flow is reduced, the equation contains a singularity within the range of radius and a meaningful solution does not exist. This result is interpreted as the limit to continuous operation and reasonable agreement between this predicted limit and the appearance of abrupt stall (experimentally) is found.     

Quasi-steady prediction of coupled bending–torsion flutter under rotating stall

A method is presented in this paper to predict cascade flutter under subsonic stalled flow condition in a quasi-steady manner. The ability to predict the occurrence of aeroelastic flutter is highly important from the compressor design point of view. In the present work, the well known Moore–Greitzer compression system model is used to evaluate the flow under rotating stall and the linearized aerodynamic theory of Whitehead is used to estimate the blade loading. The cascade stability is then predicted by solving the structural model, which is posed as a complex eigenvalue problem. The possibility of occurrence of flutter in both bending and torsional modes is considered and the latter is found to be the dominant one, under subsonic stalled flow, for a large range of frequency ratios examined. It is also shown that the design of compressor blades at frequency ratios close to unity may result in rapid initiation of torsional flutter in the presence of stalled flow. A frequency ratio of 0.9 is primarily emphasized for most part of the study as many interesting features are revealed and the results are physically interpreted. Roughly a pitchfork pattern of energy distribution appears to occur between bending mode and torsional mode which ensures that only one flutter mode is possible at any instant in time. A bifurcation from bending flutter to torsional flutter is shown to occur during which the frequency of the two vibrating modes appear to coalesce for a very short period of time.     

Effect of Membrane Preparation Method on Performance of Polyol Supported Membrane Used for Separation of Phenol

With increasing demands on the environment and on natural resources, there is a growing need to develop practical technologies that not only can remediate waste streams but also recover valuable components from these effluents. Membranes and membrane-based processes have attained technical and commercial importance with respect to their industrial and environmental applications. In the present paper, studies on stability of supported liquid membranes (SLMs) is reported. It has been shown that the method of preparation for SLM has an influence on the stability and lifetime of the SLM. Membranes prepared with ‘dry’ outer surfaces, free from organic wetting, were found to be more stable than the conventional SLM prepared with external surfaces wetted with a film of the organic membrane liquid phase. For phenol transport the ‘dry’ surface SLM had a similar initial flux to the ‘wet’ surface SLM, and about 2 times the flux after 50 h. Over a 50 h period the ‘dry’ SLM lost about 10% of its membrane liquid, whereas the ‘wet’ SLM lost about 45%. The difference is attributed to the loss of membrane liquid by emulsion formation at one of the aqueous-organic interfaces which would be greater for the ‘wet’ SLM with a continuous liquid film over the surface of the support.     

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures

The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.     

Turbulent flows undergoing distortion and rotation A report on EUROMECH 288

The 288th EUROMECH Colloquium, on the effects of distortion and rotation on turbulent motion, was held in Ecully (Ecole Centrale de Lyon) from 6–8 April 1992 with the author acting as chairman.

There were fifty-five participants with widely different backgrounds. A close mutual interaction did occur between scientists interested in ‘Rapid Distortion Theory’ (RDT) or linear stability analysis, non-linear stability analysis, Direct (or Large Eddy) Numerical Simulation (DNS), and one- and two-point statistical approaches. Qualitative comparisons with recent experimental results were also made. Approaches in geophyiscal turbulence were compared to those used in engineering, in order to clearly display similarities and differences in conclusions despite their different terminology and ‘jargon’.

In addition to qualitative results on the stability and turbulence, threshold values were proposed for dimensionless numbers, close to specific Rossby numbers. Several talks illustrated how the statistical approach reflected the emergence of oriented structures. New proposals for improving one- and two-point closure models for rotational mean flows were introduced.

Twenty-seven papers were presented, covering these different topics. These papers are summarized in this report with the purpose of giving an up-to-date view of current research in rotating fluids with distortion.     

On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308, 2014) on the well-posedness theory for the three-dimensional Prandtl equations with a special structure.     

An experimental determination of heat transfer near the leading edge of a film-cooled gas turbine blade

This paper reports an experimental procedure for the determination of the local heat transfer coefficient in the vicinity of the leading edge of a film-cooled gas turbine blade. By invoking the heat-mass transfer analogy, and measuring the sublimation rate of naphthalene, the influence of film coolant ejection on the magnitude of the local transport coefficients is determined. A novel apparatus and experimental technique have been designed to eliminate the need to know the physical properties of the naphthalene. Results of experiments using a ‘relative mass transfer method’ for a blowing rate (injection velocity to mainstream velocity ratio) from 0·2 to 1·0 and a mainstream Reynolds number of 3·48 × 10⁴ are presented and compared with previous heat transfer data.     

Numerical investigation of dynamic stall of an oscillating aerofoil

The flow structure around an NACA 0012 aerofoil oscillating in pitch around the quarter-chord is numerically investigated by solving the two-dimensional compressible N–S equations using a special matrix-splitting scheme. This scheme is of second-order accuracy in time and space and is computationally more efficient than the conventional flux-splitting scheme. A ‘rigid’ C-grid with 149 × 51 points is used for the computation of unsteady flow. The freestream Mach number varies from 0.2 to 06 and the Reynolds number from 5000 to 20,000. The reduced frequency equals 0.25–0.5. The basic flow structure of dynamic stall is described and the Reynolds number effect on dynamic stall is briefly discussed. The influence of the compressibility on dynamic stall is analysed in detail. Numerical results show that there is a significant influence of the compressibility on the formation and convection of the dynamic stall vortex. There is a certain influence of the Reynolds number on the flow structure. The average convection velocity of the dynamic stall vortex is approximately 0.348 times the freestream velocity.     

The mechanics of soil cutting by a rotating wire

The cutting of soil by a rotating wire analogous to the tip of a rotary tiller blade while cutting a two-dimensional soil slice over a range of ‘fetch-ratios’ (bite length/depth-ratios) in a quasi-static condition is presented. A theoretical models based on Mohr-Coloumb soil mechanics has been proposed to predict forces on the wire (tip). The model is dependent upon observed passive general shear failure of the soil slice towards the curved free surface of a previous cut and the lateral local shear failure towards the undeformed soil. The predicted forces in a frictional soil and in a pure cohesive medium (artificial clay) agreed well with experimental results.     

Dynamic stability of a propagating crack

In this work we investigate the stability of a straight two-dimensional dynamically propagating crack to small perturbation of its path. Willis and Movchan (J. Mech. Phys. Solids 43 (1995) 319; J. Mech. Phys. Solids 45 (1997) 591) constructed formulae for the perturbations of the stress intensity factors induced by a small three-dimensional dynamic perturbation of a nominally plane crack. Their solution is exploited here to derive equations for the in-plane and out-of-plane perturbations of the crack path making use of the Griffith fracture criterion and the principle of “local symmetry” (i.e the crack propagates so that local K_II=0). We consider a crack propagating in a body loaded by a pair of point body forces and subjected to a remote uniaxial stress, aligned with the direction of the unperturbed crack. We assume that the loading follows the crack as the crack advances and is such that the unperturbed crack is subjected to Mode I loading. We perform an analysis of the stability of the dynamic crack in a similar way as in earlier work (Obrezanova et al., J. Mech. Phys. Solids 50 (2002) 57) on the quasistatically advancing crack. We present numerical results illustrating the influence of the crack velocity on the crack stability. Numerical computations of the possible crack paths have been performed which show that at velocities of crack propagation exceeding about one-third of the speed of Rayleigh waves the crack may admit one or more oscillatory modes of instability.