A THEORETICAL MODEL FOR NONLINEAR ORBITAL MOTIONS OF ROTORS UNDER FLUID CONFINEMENT

In previous papers, Antunes and co-workers developed a theoretical model for nonlinear planar motions—motions X (t) taking place in one single direction—of rotors under fluid confinement using simplified flow equations on the gap-averaged fluctuating quantities. The nonlinear solution obtained was shown to be consistent with a linearized solution for the same problem. Also, it displayed an encouraging qualitative agreement between the nonlinear theory and preliminary experimental results. Following a similar approach, the nonlinear theoretical model is here extended to cope with orbital rotor motions—motions X (t) and Y (t) taking place in two different orthogonal directions—by developing an exact formulation for the two- dimensional dynamic flow forces. Numerical simulations of the nonlinear rotor–flow coupled system are presented and compared with the linearized model. These yield similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative dynamics stemming from the linearized and nonlinear models may be quite distinct. Preliminary experimental results also indicate that the nonlinear flow model leads to better predictions of the rotor dynamics when the eccentricity is significant, when approaching instability, and for linearly unstable regimes.     

A consideration on pre- and post-instability of an axisymmetric elastic beam subjected to axial leakage flow

The dynamical behavior of an axisymmetric elastic beam subjected to axial leakage flow is investigated numerically and experimentally. The coupled equations of motion for a fluid and a beam structure are derived using the Navier–Stokes equation for an axial leakage flow-path and the Euler–Bernoulli beam theory. Performing complex eigenvalue analysis, the variation of the dynamic behavior during pre- and post-instability is investigated with respect to increasing axial leakage flow velocity. Also, an experiment was performed to determine the critical velocity of the unstable dynamic behavior of an axisymmetric elastic beam confined in a concentric cylinder subjected to axial leakage flow through a small annulus, and to measure the variation of the dynamic behavior on pre- and post-instability when the unstable phenomenon with the lower predominant frequency is shifted to the higher one. The relationships between the axial flow velocities and the unstable phenomena are clarified for the transition from the lower mode to the higher mode by comparing the theoretical calculations with experimental observations. Especially, the generation of traveling waves and the energy balance for the distortion of vibration response in the axial direction are discussed and considered at the transition region of the complex coupled vibration response of an axisymmetric elastic beam subjected to an axial leakage flow. Numerical and experimental results are found to be in quite good agreement.     

DYNAMICS OF ROTORS IMMERSED IN ECCENTRIC ANNULAR FLOW. PART 1: THEORY

A theoretical model is developed for predicting the dynamic behaviour and stability of a rotating shaft immersed in both a concentric or eccentric fluid annulus. Of specific concern here are geometries presenting a moderate flow-confinement, for which scarce results exist. Theoretical results show that moderate gap configurations lead to quite distinct flow forces, when compared to typical bearing configurations. Also, the annulus eccentricity is shown to be a very important parameter, controlling rotor dynamics and stability. In a companion paper (Part 2), experimental results are reported and favourably compared with the present theory.     

A boundary perturbation solution for the hydrodynamic stability of blood flow in a conetype blood processor

This paper may be the first attempt to solve the flow-field ofhomogeneous fluid between concentric cones(narrow gap)withequal angular velocities of rotation by using a boundary per-turbation method,and the hydrodynamic stability of blood flowin the cone-type blood processor is verified,under conditionsof the narrow gap between the cones and small axial Re numbers.This paper also verifies the hydrodynamic stability betweenconcentric cylinders(narrow gap)with equal angular veloci-ties of rotation by using a new mathematical technic.These theoretical analyses are in agreement with the ex-perimental observations,which have been made by Shanghai Me-dical Instruments Institute.     

温度场作用下输流悬臂碳纳米管的颤振失稳分析

以非局部弹性理论为基础,采用欧拉-伯努利梁模型,考虑碳纳米管的小尺度效应,应用哈密顿原理获得了温度场作用下的输流悬臂单层碳纳米管（SWCNT）的振动控制方程以及边界条件,依靠微分变换法（DTM法）对此高阶偏微分方程进行求解,通过数值计算研究了温度场中悬臂单层输流碳纳米管的振动与颤振失稳问题。结果表明：管内流体流速、温度场中温度变化情况与小尺度参数都会对系统振动频率以及颤振失稳临界流速产生影响。其中,小尺度效应将会降低悬臂输流系统的稳定性,使系统更为柔软;而高温场与低温场对系统动态失稳的影响不同,低温场中随温度变化值的增加,系统的稳定性提高;高温场这一作用效果恰好与之相反。     

Numerical and experimental analysis of the first-and second-mode instability in a rotor-bearing system

Aiming at the oil film instability of the sliding bearing at high speeds, a rotor test rig is built to study the non-linear dynamic behaviours caused by the first- and second-mode instability. A lumped mass model (LMM) of the rotor system considering the gyroscopic effect is established, in which the graphite self-lubricating bearing and the sliding bearing are simulated by a spring–damping model and a nonlinear oil film force model based on the assumption of short bearings, respectively. Moreover, a finite element model is also established to verify the validity of the LMM. The researches focus on the effects of two loading conditions (the first- and second-mode imbalance excitation) on the onset of instability and nonlinear responses of the rotor-bearing system by using the amplitude–frequency response, spectrum cascade, vibration waveform, orbit, and Poincaré map. Finally, experiments are carried out on the test rig. Simulation and experiment all show that oil film instability can excite complicated combination frequency components about the rotating frequency and the first-/second-mode whirl/whip frequency.     

Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array

Fluidelastic instability is considered a critical flow induced vibration mechanism in tube and shell heat exchangers. It is believed that a finite time lag between tube vibration and fluid response is essential to predict the phenomenon. However, the physical nature of this time lag is not fully understood. This paper presents a fundamental study of this time delay using a parallel triangular tube array with a pitch ratio of 1.54. A computational fluid dynamics (CFD) model was developed and validated experimentally in an attempt to investigate the interaction between tube vibrations and flow perturbations at lower reduced velocities U_r=1–6 and Reynolds numbers Re=2000–12 000. The numerical predictions of the phase lag are in reasonable agreement with the experimental measurements for the range of reduced velocities U_g/fd=6–7. It was found that there are two propagation mechanisms; the first is associated with the acoustic wave propagation at low reduced velocities, U_r<2, and the second mechanism for higher reduced velocities is associated with the vorticity shedding and convection. An empirical model of the two mechanisms is developed and the phase lag predictions are in reasonable agreement with the experimental and numerical measurements. The developed phase lag model is then coupled with the semi-analytical model of Lever and Weaver to predict the fluidelastic stability threshold. Improved predictions of the stability boundaries for the parallel triangular array were achieved. In addition, the present study has explained why fluidelastic instability does not occur below some threshold reduced velocity.     

THE FLOW-INDUCED VIBRATION OF A FLEXIBLE STRIP HANGING VERTICALLY IN A PARALLEL FLOW PART 1: TEMPORAL AEROELASTIC INSTABILITY

The effect of several parameters in a fluid–strip system are studied for linear/nonlinear models in detail. Such parameters are: the number of modes in the Galerkin discretization, the length of the strip, and the flow velocity. The present simulation clearly shows that when nonlinear forces are considered the response approaches a flutter-type limit cycle at supercritical flow speeds. The Reynolds number at which the strip begins to oscillate is about 10⁴–10⁵. With a further increase of the flow velocity the strip oscillates regularly. At higher flow velocities the oscillation becomes violent and irregular. The amplitude, frequency and drag coefficient at the limit cycle are presented as functions of the flow velocity for a given strip. The numerical predictions are in qualitative agreement with previous experimental data.     

Fully suspended, five-axis, three-magnetic-bearing dynamic spin rig with forced excitation

A significant advancement in the dynamic spin rig (DSR), i.e., the five-axis, three-magnetic-bearing DSR, is used to perform vibration tests of turbomachinery blades and components under rotating and non-rotating conditions in a vacuum. The rig has three magnetic bearings as its critical components: two heteropolar radial active magnetic bearings and a magnetic thrust bearing. The bearing configuration allows full vertical rotor magnetic suspension along with a feedforward control feature, which enables the excitation of various modes of vibration in the bladed disk test articles. The theoretical, mechanical, electrical, and electronic aspects of the rig are discussed. Also presented are the forced-excitation results of a fully levitated, rotating and non-rotating, unbladed rotor and a fully levitated, rotating and non-rotating, bladed rotor in which a pair of blades were arranged 180° apart from each other. These tests include the “bounce” mode excitation of the rotor in which the rotor was excited at the blade natural frequency of 144 Hz. The rotor natural mode frequency of 355 Hz was discerned from the plot of acceleration versus frequency. For non-rotating blades, a blade-tip excitation amplitude of approximately 100 g A⁻¹ was achieved at the first-bending critical (≈144 Hz) and at the first-torsional and second-bending blade modes. A blade-tip displacement of 1.778×10⁻³m (70 mils) was achieved at the first-bending critical by exciting the blades at a forced-excitation phase angle of 90° relative to the vertical plane containing the blades while simultaneously rotating the shaft at 3000 rpm.     

Oscillatory rotationally symmetric loss of stability of nonisothermal couette flow

A study is made of the stability of nonisothermal Couette flow — steady flow of a viscous heat conducting fluid between two rotating concentric cylinders heated to different temperatures. The methods of perturbation theory are used to establish conditions sufficient for bifurcation of a neutral curve of oscillatory instability from the neutral curve of monotonic instability. Computer calculations show that for certain values of the parameters of the problem these conditions are realized and there is an oscillatory loss of stability of the nonisothermal Couette flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–80, January–February, 1984.I thank V. I. Yudovich for constant interest in the work.     

Loss Reduction by Means of Two-Dimensional Roughness Elements on the Suction Surface of a Linear Turbine Rotor Cascade

This paper reports the results of experimental investigations carried out to reduce pressure losses by means of two-dimensional roughness elements (in the form of stainless steel tubes of different diameters). The roughness elements are fixed at various axial stations on the suction surface of 120° turning, 175 mm chord, impulse turbine rotor blades. Flow measurements are carried out at the exit of the cascade at five axial stations, using a five-hole probe operating in non-nulling mode. In addition, the blade surface static pressure distribution is measured. The data from the five-hole probe measurements are used to calculate pressure, velocity and flow angle distributions at the cascade exit and these results are used to calculate mass averaged values and integral parameters such as wake half-width, loss coefficient, etc. The static pressure distribution is altered very little except near the roughness element. The lift coefficient remains almost constant for all configurations and the drag coefficient is reduced for some configurations. The non-dimensional total pressure defects in the wake for all configurations followed Gaussian distribution. A two-dimensional roughness element of 0.6 mm diameter placed at 0.65 chord on the suction surface showed an appreciable reduction in pressure losses.     

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.     

On the Stability of a Laminar Wall Jet with Heat Transfer

The hydrodynamic stability of a low speed, plane, non-isothermal laminar wall jet at a constant temperature boundary condition was investigated theoretically and experimentally. The mean velocity and temperature profiles used in the stability analysis were obtained by implementing the Illingworth–Stewartson transformation that allows one to extend the classical Glauert solution to a thermally non-uniform flow. The stability calculations showed that the two unstable eigenmodes coexisting at moderate Reynolds numbers are significantly affected by the heat transfer. Heating is destabilizing the flow while cooling is stabilizing it. However, the large-scale instabilities associated with the inflection point of the velocity profile still amplify in spite of the high level of the stabilizing temperature difference. The calculated stability characteristics of the wall jet with heat transfer were compared with experimental data. The comparison showed excellent agreement for small amplitudes of the imposed perturbations. The agreement is less good for the phase velocities of the sub-harmonic wave and this is attributed to experimental difficulties and to nonlinear effects.     

Galloping instabilities of two-dimensional triangular cross-section bodies

Galloping is a type of aeroelastic instability characterized by large amplitude, low frequency, normal to wind oscillations. It normally appears in bodies with small stiffness and structural damping when they are placed in a flow and the incident velocity is high enough. In this paper a systematic approach for the analysis of galloping of triangular cross-section bodies is reported. Wind tunnel experiments have been conducted aiming at establishing the unstable characteristics of isosceles triangular cross-section bodies when subjected to a uniform flow with angles of attack ranging from 0 to 180°. The results have been summarized in a stability map, where galloping instability zones in the angle of attack—main vertex angle plane—are identified.     

Thermoconvective instability of longitudinal flow in a vertical layer

An experimental study has been made of convection in a vertical slit cavity heated from below and with longitudinal horizontal forced flow. It was shown that the convective stability of such flow increases appreciably when the velocity of the forced flow is raised. In the case of slow pumping, an increase in the pressure difference leads to superposition on the rectilinear flow of first monotonic convection and then auto-oscillatory convection. At high flow velocities, the instability is immediately of an oscillatory nature. A diagram of the flow regimes is constructed, and the evolution of the supercritical structures described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhikosti i Gaza, No. 2, pp. 28–33, March–April, 1984.     

A computational investigation of unsteady turbulent wake–boundary-layer interaction

The relative motion of rotor and stator blade rows in a turbomachine generates periodically unsteady flow on the blades due to travelling wake perturbations. To better understand the attendant wake–boundary-layer interaction a calculation procedure was developed to model the behaviour of this complex unsteady flow. Due to nonlinear interactions with the boundary layer, the travelling discrete frequency wakes were found to decrease the velocity profile shape factor. For the range of reduced frequencies examined (=0.33–9.33) the skin-friction coefficient was found to be frequency dependent. The calculated results for both steady and unsteady velocity profiles, and for skin friction compared well with experimental data. Although the agreement between measured and calculated velocity phase shift was poor, in both experimental and model results the negative phase shift throughout the boundary layer due to the travelling-wave fluctuations has been captured.     

Studies on the law of laminar helical flow of power-law fluid in eccentric annuli

According to the principle of fluid mechanics,the law of laminar,helical flow of power-law fluid in eccentric annuli is studied extensively in this paper.The apparent viscosity,velocities distribution of laminar helical flow of power-law fluid are discussed and calculating methods of flow rate and pressure loss are presented.The factors influencing flow are also analysed.On the basis of theoretical studies some new results of the present paper are compared with the theories of the helical flow of the power-law fluid in concentric annuli.The test verified that theoretical formulas in this article are proper and general.They can be used for calculating hydraulic parameters in drilling engineering.     

Aerodynamic control of bridge cables through shape modification: A preliminary study

This paper examines the viability of modifying bridge cable shape and surface for the purpose of controlling wind-induced vibrations. To this end, an extensive wind-tunnel test campaign was carried out on various cable shapes about the critical Reynolds number region. Cable shapes were chosen to passively modify the flow in a particular manner. Tested shapes included those which have some form of waviness, faceting and shrouding. Section models were tested using a static inclined rig, allowing them to be installed at yawed cable-wind angles for both smooth and turbulent flow conditions. The aerodynamic damping of the tested cylinders is evaluated by applying both 1- and 2-dof quasi-steady aerodynamic instability models. This allows for the prediction of regions of aerodynamic instability, as a function of flow angle and Reynolds number. Whilst the plain, wavy and faceted cylinders are predicted to suffer from either dry inclined galloping, “drag crisis” or Den Hartog galloping, the shrouded cylinder is found to be stable for all angles of attack, albeit with an increase in drag at typical design wind velocities. Finally, turbulent flow is found to introduce an increased amount of aerodynamic damping mainly by providing a more constant lift force over tested Reynolds numbers.     

Effect of transverse stream velocity in an incompressible boundary layer on the stability of the laminar flow regime

The stability of the laminar flow regime in the boundary layer developed on a wall is increased considerably by the relatively slight extraction of fluid from the wall [1–4]. In the theoretical study of this phenomenon, all the investigators known to the present authors have taken into account only the increase in the fullness of the velocity profile in the boundary layer with suction. Computations of the stability characteristics have been made on the assumption that there are no transverse velocities in the laminar boundary layer.We present below an analysis of the stability of the laminar boundary layer in the presence of a constant transverse velocity in the near-wall region (suction). The calculations made show the existence for a given velocity profile in the boundary layer of a relative suction velocity v=v such that with suction velocities greater than v the flow remains stable at all Reynolds numbers, while the method used in the cited references gives a definite finite critical Reynolds number, equal in our notation to the Reynolds number at v=0, for each relative suction velocity.It was found that with suction of fluid from the boundary layer the region of instability has finite dimensions, i.e., there exist lower and upper critical Reynolds numbers. The flow is stable if its Reynolds number is less than the lower, or greater than the upper values of the critical Reynolds number.The instability region diminishes with increase in the relative suction velocity, and at a value of this velocity which is specific for each value of the velocity profile the instability region degenerates into a point-the flow becomes absolutely stable. Thus, with distributed suction it is advisable to increase the relative suction velocity only to a definite magnitude corresponding to disappearance of the instability region. The computational results presented make it possible to estimate this velocity for velocity profiles ranging from a Blasius profile to an asymptotic profile. Specific calculations were made for a family of Wuest profiles, since under actual conditions with suction there always exists a starting segment of the boundary layer [1, 2].     

A case of strong nonlinearity: Intermittency in highly turbulent flows

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution to the Euler equations for inviscid fluids. Then we show that single-point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions to the fluid equations. Conversely, those experimental velocity–acceleration correlations are contradictory to the Kolmogorov scaling laws.