Study of a non-linear mechanical system using Boubaker polynomials expansion scheme BPES

Differential equations governing mechanical system behaviours have to be transformed into algebraic equations using the appropriate analytical and numerical tools. This study is concerned with the identification of a non-linear 2-degree-of-freedom mechanical system using the Boubaker polynomials expansion scheme (BPES). Solutions are plotted in the frequency-energy plane and are compared to other results published so far.     

Highly spatially resolving laser Doppler velocity measurements of the tip clearance flow inside a hard disk drive model

The flow in the tip clearance of a hard disk drive model has been investigated with laser Doppler techniques. The flow was driven by co-rotating disks inside a cylindrical enclosure in order to simulate a hard disk drive used for data storage devices. The main focus of the investigation was on the understanding of complex flow behavior in the narrow gap region between the disk tip and the outer shroud wall, which is supposed to be one of the causes of flow induced vibration of the disks. Experiments in the past have never been able to examine this region because of the lack of the spatial resolution of sensors in the highly three-dimensional flow in the region. In the present investigation, the flow velocity in the tip clearance region was measured with optical measurement techniques for the first time. The flow behaviors are investigated for four different conditions with two different gap widths and two different shapes of the shroud walls with and without ribs. The velocity measurements were taken both with conventional laser Doppler velocimetry and using a laser Doppler velocity profile sensor with a spatial resolution in the micrometer range. The circumferential velocity component was measured along the axial and radial directions. The steep gradients of the circumferential mean velocity in both directions were successfully captured with a high spatial resolution, which was achieved by the velocity profile sensor. From the supplementary investigations, the existence of vortex structures in the tip clearance region was confirmed with a dependence on the shroud gap width and the shroud shape. The interactions of the two boundary layers seem to be the source of the complex three-dimensional behaviors of the flow in this region.     

Modal expansion of the perturbation velocity potential for a cantilevered fluid-conveying cylindrical shell

The subject of this paper is the application of the method of separation of variables and the Galerkin method for discretization of the equations of motion for a cantilevered cylindrical fluid-conveying shell. The perturbation velocity potential is expressed in terms of a series of orthonormal beam modal functions. The final Galerkin generalized fluid force coefficients are simple, compact, and easy to evaluate numerically. To validate the method, comparisons with results obtained from the Fourier transform method are made. Mismatch between the actual axial fluid modes and the assumed modes affects the Galerkin coefficients to some extent, but the unstable eigenvalue branch is only affected slightly over a wide range of system parameters, and critical flow speeds predicted by the two methods generally agree well.     

Effect of gas expansion on the velocity of a Taylor bubble: PIV measurements

The effect of gas expansion on the velocity of a Taylor bubble was studied experimentally. The velocity field in the liquid ahead of a Taylor bubble was measured by particle image velocimetry (PIV), and the bubble velocity was measured with two pairs of laser diodes and photocells. The experiments were done in a 7.0 m long vertical tube with a 32 mm internal diameter. Solutions of carboxymethylcellulose (CMC) polymer with weight percentages between 0.01% and 0.1% were used. The expansion of slug gas induces an increase in the bubble velocity and a corresponding displacement of the liquid ahead of the bubble. The velocity of the bubble increases by an amount equal to the maximum velocity in the liquid displaced. For the solutions studied, the induced velocity profile was parabolic and the bubble velocity increase was equal to the liquid velocity at the tube axis, i.e., twice the mean velocity in the liquid displaced. The corrected velocity obtained by subtracting the velocity increase from the value of the bubble velocity is independent of the bubble length.     

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples:turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder (GL) model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.     

Numerical solution of the two‐phase expansion of a metastable flashing liquid jet using the dispersion‐controlled dissipative scheme

This paper presents a study of the stationary phenomenon of superheated or metastable liquid jets, flashing into a two‐dimensional axisymmetric domain, while in the two‐phase region. In general, the phenomenon starts off when a high‐pressure, high‐temperature liquid jet emerges from a small nozzle or orifice expanding into a low‐pressure chamber, below its saturation pressure taken at the injection temperature. As the process evolves, crossing the saturation curve, one observes that the fluid remains in the liquid phase reaching a superheated condition. Then, the liquid undergoes an abrupt phase change by means of an oblique evaporation wave. Across this phase change the superheated liquid becomes a two‐phase high‐speed mixture in various directions, expanding to supersonic velocities. In order to reach the downstream pressure, the supersonic fluid continues to expand, crossing a complex bow shock wave. The balance equations that govern the phenomenon are mass conservation, momentum conservation, and energy conservation, plus an equation‐of‐state for the substance. A false‐transient model is implemented using the shock capturing scheme: dispersion‐controlled dissipative (DCD), which was used to calculate the flow conditions as the steady‐state condition is reached. Numerical results with computational code DCD‐2D v1 have been analyzed. Copyright © 2009 John Wiley & Sons, Ltd.     

Construction of the velocity potential of a semiinfinite liquid column excited by a spherical calotte oscillating inside a cylindrical cavity

A semiinfinite liquid column excited by a spherical calotte oscillating inside a rigid cylindrical cavity is considered. The solution of the Laplace equation is searched for under boundary conditions on spherical, cylindrical, and flat surfaces bounding the liquid. The possibility of decomposing a spherical harmonic function in terms of cylindrical harmonic functions and vice versa is considered. The potential constructed is used to evaluate the pressure and the velocity of the liquid. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 81–89, March, 2000.     

Mean velocity and moments of turbulent velocity fluctuations in the wake of a model ship propulsor

Pod drives are modern outboard ship propulsion systems with a motor encapsulated in a watertight pod, whose shaft is connected directly to one or two propellers. The whole unit hangs from the stern of the ship and rotates azimuthally, thus providing thrust and steering without the need of a rudder. Force/momentum and phase-resolved laser Doppler anemometry (LDA) measurements were performed for in line co-rotating and contra-rotating propellers pod drive models. The measurements permitted to characterize these ship propulsion systems in terms of their hydrodynamic characteristics. The torque delivered to the propellers and the thrust of the system were measured for different operation conditions of the propellers. These measurements lead to the hydrodynamic optimization of the ship propulsion system. The parameters under focus revealed the influence of distance between propeller planes, propeller frequency of rotation ratio and type of propellers (co- or contra-rotating) on the overall efficiency of the system. Two of the ship propulsion systems under consideration were chosen, based on their hydrodynamic characteristics, for a detailed study of the swirling wake flow by means of laser Doppler anemometry. A two-component laser Doppler system was employed for the velocity measurements. A light barrier mounted on the axle of the rear propeller motor supplied a TTL signal to mark the beginning of each period, thus providing angle information for the LDA measurements. Measurements were conducted for four axial positions in the slipstream of the pod drive models. The results show that the wake of contra-rotating propeller is more homogeneous than when they co-rotate. In agreement with the results of the force/momentum measurements and with hypotheses put forward in the literature (see e.g. Poehls in Entwurfsgrundlagen für Schraubenpropeller, 1984; Schneekluth in Hydromechanik zum Schiffsentwurf, 1988; Breslin and Andersen in Hydrodynamics of ship propellers, 1996; Schneekluth and Bertram in Ship design for efficiency and economy, 1998), the co-rotating propellers model showed a much stronger swirl in the wake of the propulsor. The anisotropy of turbulence was analyzed using the anisotropy tensor introduced by Lumley and Newman (J Fluid Mech 82(1):161–178, 1977). The invariants of the anisotropy tensor of the wake flow were computed and were plotted in the Lumley–Newman-diagram. These measurements revealed that the anisotropy tensor in the wake of ship propellers is located near to the borders of the invariant map, showing a large degree of anisotropy. They will be presented and will be discussed with respect to applications of turbulence models to predict swirling flows.     

Investigation of the velocity field in a full-scale model of a cerebral aneurysm

Due to improved and now widely used imaging methods in clinical surgery practise, detection of unruptured cerebral aneurysms becomes more and more frequent. For the selection and development of a low-risk and highly effective treatment option, the understanding of the involved hemodynamic mechanisms is of great importance. Computational Fluid Dynamics (CFD), in vivo angiographic imaging and in situ experimental investigations of flow behaviour are powerful tools which could deliver the needed information. Hence, the aim of this contribution is to experimentally characterise the flow in a full-scale phantom model of a realistic cerebral aneurysm. The acquired experimental data will then be used for a quantitative validation of companion numerical simulations. The experimental methodology relies on the large-field velocimetry technique PTV (Particle Tracking Velocimetry), processing high speed images of fluorescent tracer particles added to the flow of a blood-mimicking fluid.First, time-resolved planar PTV images were recorded at 4500 fps and processed by a complex, in-house algorithm. The resulting trajectories are used to identify Lagrangian flow structures, vortices and recirculation zones in two-dimensional measurement slices within the aneurysm sac. The instantaneous inlet velocity distribution, needed as boundary condition for the numerical simulations, has been measured with the same technique but using a higher frame rate of 20,000 fps in order to avoid ambiguous particle assignment. From this velocity distribution, the time-resolved volume flow rate has been also derived. In this manner, a direct comparison between numerical simulations and PTV measurements will be possible in the near future, opening the door for highly accurate computational predictions.     

Estimation of the angular rotation velocity of a body using a tracking system

The problem of finding the angular rotation velocity of a body using the orientation matrix whose elements are determined by a tracking system is considered. Two methods of solving this problem are compared. One of them is based on the representation of the function in the form of a partial sum of the Fourier series and the second one is based on using the Savitzky-Golay filter.     

Computation of three-dimensional flow using the Euler equations and a multiple-grid scheme

The unsteady Euler equations are numerically solved using the finite volume one-step scheme recently developed by Ron-Ho Ni. The multiple-grid procedure of Ni is also implemented. The flows are assumed to be homo-enthalpic; the energy equation is eliminated and the static pressure is determined by the steady Bernoulli equation; a local time-step technique is used. Inflow and outflow boundaries are treated with the compatibility relations method of ONERA. The efficiency of the multiple-grid scheme is demonstrated by a two-dimensional calculation (transonic flow past the NACA 12 aerofoil) and also by a three-dimensional one (transonic lifting flow past the M6 wing). The third application presented shows the ability of the method to compute the vortical flow around a delta wing with leading-edge separation. No condition is applied at the leading-edge; the vortex sheets are captured in the same sense as shock waves. Results indicate that the Euler equations method is well suited for the prediction of flows with shock waves and contact discontinuities, the multiple-grid procedure allowing a substantial reduction of the computational time.     

A predictive model based on a 3-D computational approach for film cooling effectiveness over a flat plate using GMDH-type neural networks

A 3-D numerical solution is implemented for investigating incompressible turbulent flow and thermal fields of film cooling through a single row of inclined cylindrical holes over a flat plate model. The effects of parameters of interest on the film cooling performance have been simulated. The group method of data handling (GMDH)-type neural networks successfully employed for modeling and presenting a correlation for area-weighted average adiabatic film cooling effectiveness.     

Trajectories of a ball moving inside a spherical cavity using first integrals of the governing nonlinear system

Nonlinear Dynamics - Analytical study of ball vibration absorber behavior is presented in the paper. The dynamics of trajectories of a heavy ball moving without slipping inside a spherical cavity...     

Asymptotic model of the development of separations inside a boundary layer under the action of a traveling pressure wave

A model of the nonlinear interaction between a pressure perturbation traveling at a constant velocity and an incompressible boundary layer is constructed when its near-wall part is described by the “inviscid boundary layer” equations. A steady-state solution is managed to obtain in the finite form under the assumption that it exists in a moving coordinate system. It is shown that the boundary layer can easily overcome pressure perturbations whose amplitude is not higher than the dynamic pressure calculated from the velocity of the pressure perturbation. At the higher pressure perturbation amplitudes a vortex sheet sheds from the body surface to the boundary layer. The vortex sheet represents an unstable surface of the tangential discontinuity which separates the regions of the direct and reverse separation flows. In the case of an arbitrary shape of the pressure perturbation the surface of the tangential discontinuity sheds from the body surface at a finite angle with the formation of a stagnation point. An example of the pressure perturbation in which the vortex sheet sheds from the body surface along the tangent is constructed.     

Measurement of the velocity of a rarefied gas flow using an ionic tag

The article expounds a method for measuring the velocity in a flow close to free-molecular from the motion of an ion tag, formed using an electron beam. It examines the effect of the field of the beam and of the inherent volumetric charge of the tag on the results of the measurements; the conditions under which this effect can be neglected are calculated. The velocity In highly rarefied jets of nitrogen and argon was measured. The method of an ion tag, for example [1, 2], consists in the fact that the gas is ionized (tagged) using a narrow pulsed beam of electrons, and the velocity of the gas is determined from the time of flight of the tag from a known base. The tag consists in an ion cloud, since the electrons forming with ionization of the gas have an energy of several electron volts and leave the flow after a time t_e which is small in comparison with the time of the motion of the ions t_i (t_e<10^?2t_i). The presence of the inherent volumetric charge of the ion cloud leads to washing-out of the ion tag and to its additional acceleration. With an increase in the degree of rarefaction, particularly in a free-molecular flow, where the ions move without collisions with the molecules of the flow, the effect of the inherent fields increases, and the velocity of the tag may differ considerably from the velocity of the neutral gas. Insufficient attention has been paid to this question in the literature.     

A regularised one-dimensional drop formation and coalescence model using a total variation diminishing (TVD) scheme on a single Eulerian grid

The breakup of an axisymmetric viscous jet is considered in the lubrication approximation. The discretised equations are solved on a fixed equidistant one-dimensional Eulerian grid. The governing equations are implemented in a conservative second order accurate total variation diminishing (TVD) scheme, preventing the numerical diffusivity. Singularities that occur at pinchoff and coalescence are regularised by a small modification on the surface tension. The modification is of the order of the spatial step Δx. This regularisation ensures that the solution of the presented numerical model converges to the exact solution of the breakup of a jet in the lubrication approximation. The results of the presented numerical model agree quantitatively with the analytical solution of the Rayleigh–Plateau instability, and with experimental results on the final stage of the Rayleigh–Plateau instability.     

An extended traffic flow model on a gradient highway with the consideration of the relative velocity

In this paper, an extended traffic flow model on a single-lane gradient highway is proposed with the consideration of the relative velocity. The stability condition is obtained by the use of linear stability analysis. It is shown that the stability of traffic flow on the gradient varies with the slope and the coefficient of the relative velocity: when the slope is constant, the stable regions increase with the increase of the coefficient of the relative velocity; when the coefficient of the relative velocity is constant, the stable regions increase with the decrease of the slope in downhill and increase with the increase of the slope in uphill. The Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations are derived to describe the triangular shock waves, soliton waves, and kink-antikink waves in the stable, metastable, and unstable region, respectively. The numerical simulation shows a good agreement with the analytical result, which shows that the traffic congestion can be suppressed by introducing the relative velocity.