Stationary Disk Research Articles

Disk sources of the Kerr and Tomimatsu-Sato spacetimes: Construction and physical properties

We construct the disk sources matched to the exact vacuum Kerr and to the two classes of Tomimatsu-Sato spacetimes. We analyze two models of the matter forming these disks. At each radius we consider either a rotating massive ring with pressure or two counter-rotating streams of particles in circular geodesic motion. Dragging effects present in such spacetimes lead either to rotation of rings or asymmetry of both streams. We demonstrate that the model of rotating rings is general enough to describe all axisymmetric stationary disk sources with vanishing radial pressure which satisfy weak energy condition, and that centrifugal effects present in the disk sources of spacetimes with large angular momentum prevent the construction of highly compact sources made of counter-rotating streams of geodesic particles. We illustrate the radial distribution of the mass inside the disks and the angular velocities of both geodesic streams.

Numerical simulation of incompressible turbulent flow in shrouded disk system with radial outflow

The flow behavior inside the shrouded disk system is of importance in appropriate design of turbomachinery cavities and turbine test cell hydraulics dynamometer. The turbulent incompressible flow is analyzed for the shrouded disk system with axial clearance. The flow core behaves as a Batchelor type structure when a weak inflow is imposed on the disk cavity. By increasing the inflow, the central core disappears and the tangential velocity distribution is changed to Stewartson type structure. The central core again reappears by increasing the Reynolds number. The moment coefficient of rotary disk depends on superimposed flow rate coefficient and dimensionless geometrical parameters. Moment coefficient increases with increasing inflow rate while the other parameters remain constant. The coefficient is reduced by increasing the Reynolds number. Moreover, it increases with both increasing rotary and stationary disks axial distance, and decreasing clearance ratio. The experimental results of a cavity with radial clearance are used to validate the accuracy of the simulation. The results of this analysis and its development can be used in the design of turbine test cell hydraulics dynamometers.

Heat Transfer Enhancement in a Radial Turbulent Sink Flow Cooling System

The steady forced convection between two stationary parallel circular disks in a radial sink flow cooling system is investigated numerically. This investigation is devoted to study the effect of swirling flow and/or grooved surface on the heat transfer and on the thermo-hydraulic parameter. A wide range of inlet Reynolds number (Re), 100 ≤ Re ≤ 105, inlet swirl ratio (S), 0 ≤ S ≤ 20, and the gap spacing ratio (G), 0.01 ≤ G ≤ 0.1 is considered in the study. The rectangular grooves are characterized by ribs with three dimensionless lengths: height (t/δ), 0.1 ≤ t/δ ≤ 0.35, the interval spacing between ribs (i/Ro), 0.025 ≤ i/Ro ≤ 0.1, and the width of rib (w/Ro), 0.025 ≤ w/Ro ≤ 0.1. The results of the heat transfer analysis indicate that the swirling flow enhances the cooling system for plain and ribbed surfaces. However, the thermal hydraulic study indicated that the swirling flow is beneficial for plain surfaces only. And the ribs are beneficial with pure radial inflow.

Static and dynamic Brazilian disk tests for mechanical characterization of annealed and chemically strengthened glass

The intent of this research is to determine the tensile strength and fracture behavior of annealed and chemically strengthened aluminosilicate glass under static and dynamic loading conditions using Brazilian disk (BD) tests. Owing to high brittleness and very small failure strain of glass, this test method is preferred to indirectly measure the tensile strength by compressive loading of cylindrical specimens diametrically instead of using direct tensile test. In this study, based on the BD principle, the static tensile strength is measured by using the universal testing machine and a modified SHPB with pulse shaper technique is used to measure the dynamic tensile strength of two types of glasses. The synchronous high-speed photography in both static and dynamic tests is used to capture the damage/crack initiation, crack pattern, and failure process of BD specimens. Also, the strain-time history was recorded from gauges directly embedded on BD specimens in both tests. The test results showed that two types of glasses under investigation are the loading rate-sensitive. In the case of annealed glass (AG), the average dynamic tensile strength was increased three times from the static strength and a substantial increase in tensile strength of chemically strengthened glass (CSG) was observed at high loading rate. The high-speed images and collected glass debris during BD tests are analyzed and discussed to explain the rate-sensitivity, fracture mode and tensile failure process of two types of glasses. The high-speed video suggested that the main central crack caused the splitting or failure in AG and in case of CSG the main central crack along with multiple cracks the splitting or failure occurred. Finally, the scanning electron microscope (SEM) images of collected glass residue of (AG) and (CSG) was also studied to illustrate the brittle fracture surface, crack pattern and deformation mechanism in two types of glasses.

Mistuning Evaluation Comparison Via As-Manufactured Models, Traveling Wave Excitation, and Compressor Rigs

As-manufactured rotors behave quite differently than nominal as-designed rotors due to small geometric and material property deviations in the rotor, referred to as mistuning. The mistuning of a 20 bladed, integrally bladed rotor (IBR) will be evaluated via analytical methods, benchtop testing, and using a rotating compressor research facility. Analytical methods consist of the development of an as-manufactured model based on geometry measurements from a high fidelity optical scanning system. Benchtop testing of the IBR is done using a traveling wave excitation (TWE) system that simulates engine order excitation in stationary bladed disks for the purpose of determining potentially high responding blades due to mistuning. The compressor research facility utilizes blade tip timing to measure the blade vibration of the IBR. The resonant response of the IBR at various modes and harmonic excitations is investigated. A comprehensive mistuning and force amplification comparison between the as-manufactured model, TWE, and the compressor rig is performed. Mistuning of each method is evaluated using three different methods. First, the tuned absorber factor (TAF), which is a metric to determine potential high responding blades, is determined for each system. Next, mistuning is analyzed by isolating individual blades both experimentally on the bench and analytically to determine the mistuning patterns. Lastly, the mistuning determined by each system will be evaluated using a reduced-order model, namely the fundamental mistuning model identification (FMM ID). It will be shown that TAF shows variability between each method providing indications TAF may not be the best approach of force amplification predictions. Basic mistuning agreements exist when isolating blades both experimentally and analytically exhibiting as-manufactured models are capable of representing full experiments. System ID methods provide a basic agreement between both the mistuning pattern and the mistuning amplification for all three methods analyzed. This ultimately shows the importance and the ability to use as-manufactured models to help increase detailed understanding of IBR's.

The role of gravity in the prediction of the circular hydraulic jump radius for high-viscosity liquids

The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$, where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.

Flow Characteristics of Curved Rotor Stator Systems Using Large Eddy Simulation

In this paper, the new idea of application of a curved rotor disk in rotor stator systems is presented and analyzed by using the large eddy simulation technique. The geometry of the examined rotor-stator system consists of a stationary flat disk (stator), a rotating curved disk (rotor) and a stationary enclosing cylinder (shroud). A hole in the center of stator allows the flow to enter the cavity, and the clearance between the shroud and rotor enables the flow to exit the cavity. Employing elliptical bumps with different geometrical parameters on the rotor disk (creating a curvature on the rotor), the rotating curved disk is parametrized. Three cavity cases (one with a flat rotor disk, another with the maximum outflow total pressure, and the third with the highest mass flow rate) are selected for LES analysis and more detailed investigation of flow and turbulence structures. The Favre-filtered governing equations for LES analysis of compressible turbulent flows are solved for all three cases. Radial and circumferential flow velocities as well as shear and normal Reynolds stresses in different cavity regions are studied. The flow in a rotor-stator cavity is simultaneously affected by the inlet flow, rotor rotation, and the bump on rotor disk. Creating a bump on rotor disk causes increase of both the radial pressure gradient and the mass flow rate of fluid that enters the rotor-stator cavity.

Two mass conjectures on axially symmetric black hole-disk systems

We analyze stationary self-gravitating disks around spinning black holes that satisfy the recently found general-relativistic Keplerian rotation law. There is a numerical evidence that the angular velocity, circumferential radius, and angular momenta yield a bound onto the asymptotic mass of the system. This bound is proven analytically in the special case of massless disks of dust in the Kerr spacetime.

Turbulence Models for Simulation of the Flow in a Rotor-Stator Cavity

Modelling of the flow in the cavities between rotor and stator in turbomachines (e.g. pumps or turbines) is a task of great interest. Correctly evaluated pressure and velocity fields enable calculation of the disk losses and therefore assessment of efficiency. It is also crucial for determination of axial thrust and thus design of the bearings. The study demonstrates abilities of various turbulence models to describe the flow in a narrow gap between rotating and stationary disks. Numerical simulations were performed in order to find out the ability of particular models to capture unstable structures appearing during specific operating conditions as well as to calculate the velocity profiles precisely. Large Eddy Simulation (LES), Scale Adaptive Simulation (SAS), Detached Eddy Simulation (DES), Reynolds stress model (RSM) and SSTk–ωmodel were used. Obtained results were also compared with experimental measurement published by Viazzo et al. [1]

Round and round it goes

We discuss the coupling of a rotating flywheel with a nonrotating flywheel. We find an expression for the tangential force between the wheels and an expression for the change in angular momentum of the system. Then, we calculate the fraction of rotational kinetic energy which is transferred from the rotating flywheel to the nonrotating flywheel. The theoretical results are compared with the experimental results in the article Mário S M N F Gomes et al, The ‘Spinning disk touches stationary disk’ problem revisited: an experimental approach (2018 Eur. J. Phys. 39 045709). We then introduce a series of initially nonrotating flywheels between the two original flywheels which are coupled and decoupled, beginning with coupling between the original rotating flywheel and the first wheel in the series and finally coupling between the last flywheel in the series with the original nonrotating flywheel. The condition for maximum energy transfer from the original rotating flywheel to the original nonrotating flywheel for a given number of flywheels will be shown to be achieved when the masses of the flywheels constitute a geometric series if all the flywheels are of the same type. If the flywheels are not of the same type the maximum is achieved when the product of the K-factors of the inertial moments and the masses constitute a geometric series.

Effective sound speed in relativistic accretion discs around Schwarzschild black holes

For low angular momentum axially symmetric accretion flow maintained in hydrostatic equilibrium along the vertical direction, the value of the Mach number at the critical points deviates from unity, resulting in the non-isomorphism of the critical and the sonic points. This introduces several undesirable complexities while analytically dealing with the stationary integral accretion solutions and the corresponding phase portraits. We propose that the introduction of an effective dynamical sound speed may resolve the issue in an elegant way. We linear perturb the full spacetime-dependent general relativistic Euler and the continuity equations governing the structure and the dynamics of accretion disc in vertical equilibrium around Schwarzschild black holes and identify the sonic metric embedded within the stationary background flow. Such metric describes the propagation of the linear acoustic perturbation inside the accretion flow. We construct the wave equation corresponding to that acoustic perturbation and find the speed of propagation of such perturbation. We finally show that the ordinary thermodynamic sound speed should be substituted by the speed of propagation of the linear acoustic wave which has been obtained through the dynamical perturbation. Such substitution will make the value of Mach number at the critical point to be equal to unity. Use of the aforementioned effective sound speed will lead to a modified stationary disc structure where the critical and the sonic points will be identical.

The abundance of satellite galaxies in the inner region of ΛCDM Milky Way sized haloes

The concordance $\Lambda$CDM cosmology predicts tens of satellite galaxies distributed in the inner region ($<40\ \lkpc$) of the Milky Way (MW), yet at most $12$ were discovered at present day, including 3 discovered very recently by Dark Energy Survey (DES) and 5 from other surveys (e.g. PanSTARRS, MagLiteS). We use $5$ ultra-high resolution simulations of MW sized dark matter haloes from the $\aqua$ project, combined with $\galf$ semi-analytical galaxy formation model, to investigate properties of the model satellite galaxy population inside $40\ \lkpc$ of MW sized haloes. On average, in each halo this model predicts about $20$ inner satellite galaxies, among them $5$ are comparable to the classic satellites in the luminosity, these are in stark contrast to the corresponding numbers in observations. We further investigate the survivability of these model inner satellites in the presence of a central stellar disk with a set of ideal simulations. These are done by re-evolving a quarter (30) of the whole $\aqua$ inner satellite galaxies (121) by including a static disk potential in addition to the MW halo. Our finding is that the additional disk completely disrupt 40 percents of these satellites and results in $14$ satellite galaxies within the $40\ \lkpc$ of each $\aqua$ at the end, in reasonably well agreement with observations.

Qualitative relationships between structure and performance of rotating zigzag bed in distillation

Rotating zigzag bed (RZB) is the centrifugal tray column, a counterpart to rotating packed bed (RPB) and has been commercialized in a number of distillation applications. In this paper, qualitative relationships between the structure and the performance of RZB rotor are studied to adjust its performance and optimize rotor structure. Several structural modifications of RZB rotor are analyzed, such as varying height of stationary baffles and blank region on the rotating baffles, additional packing layers on the stationary baffles and under the stationary disc. The performance of these RZB rotors is experimentally measured and compared to understand the fundamentals of mass transfer mechanism. The structural variations with a larger height of stationary baffles, larger height of the blank region on the rotating baffles, packing layers on the stationary baffles improves the mass transfer to some extent, however at the cost of higher pressure drop. Moreover, the stationary baffles account for the majority of the mass transfer and the lower part of baffles has a larger contribution than the upper part. Packing rings under the stationary disc contribute little to the mass transfer, which implies a very small entrainment of the liquid droplets. These qualitative relationships between the structure and the performance provide a means to improve the performance of the RZB rotor without changing the overall rotor dimensions. An RZB with an adaptable performance produced by rotor structure modifications has an extensive prospect in the industrial applications.

3D hydrodynamical simulation of accretion disk in binary star system using RKDG CFD solver

We present the results of 3D-hydrodynamical simulations of accretion disk in close binary star system. The model includes the optical star filling its Roche lobe, a gas stream emanating from the inner Lagrangian point of the binary system, and the accretion disc structure. A cold hydrogen gas stream is initially emitted towards a point-like gravitational centre. A stationary accretion disc is formed in about 5 orbital periods after the beginning of accretion. The model uses realistic cooling function for hydrogen. The simulation of accreting gas dynamics is performed using second order Runge — Kutta discontinuous Galerkin CFD solver for unstructured tetrahedral meshes. The monotonicity technique based on Hermite WENO solution reconstruction is implemented. Utilizing the hydrodynamical simulations the sinthetic light curves of the system are calculated as the volume emission of optically thin layers along the line of sight. The simulations results are in a good agreement with observations.

Electrodeposited AgPd alloy coatings as efficient catalysts for the ethanol oxidation reaction

The Pd and three AgPd alloy layers (AgPd1, AgPd2 and AgPd3) were electrodeposited onto Au disc electrodes from the solution containing high concentration of chloride ions (>12 M). All coatings were investigated by scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDS), anodic linear sweep voltammetry (ALSV), while their surface composition was investigated by X-ray photoelectron spectroscopy (XPS). The AgPd1 and AgPd2 samples were electrodeposited at different constant current densities (−0.178 mA cm−2 and -0.415 mA cm−2 respectively) to the charge of −0.2 C cm−2 (thickness ∼ 0.18 μm) at a stationary disc electrode, while the sample AgPd3 was electrodeposited to the charge of −3.0 C cm−2 (thickness ∼ 2.8 μm) at a constant current density of −7.0 mA cm−2 under the conditions of convective diffusion. Samples AgPd1 and AgPd2 had similar morphologies of low roughness, while the morphology of AgPd3 was characterized by large crystals and higher roughness. The most active and the most poisoning tolerant coatings for ethanol oxidation reaction (EOR) are the AgPd3 and AgPd1 alloy samples, containing 72.6 at.% Ag – 27.4 at.% Pd and 84.7 at.% Ag – 15.2 at.% Pd respectively (XPS analysis). In this study, we demonstrated for the first time that the activity for the EOR at AgPd alloys was closely related to the amount of non-reduced Ag2O (most probably as Ag – hydroxide). Accordingly, all AgPd alloy samples had to be cycled in the potential region of Ag2O formation and reduction before the investigation of the EOR, in order to provide their catalytic activity towards the EOR.

Investigation of flow characteristics from an inclined jet on a heated rotating disk

Impinging jet directly hits the object surface to achieve drying, heating or cooling purpose. Jet flow and rotation parameters will affect the flow field. In this paper, the flow pattern and velocity distribution of the inclined jet on a rotating disk under different experimental conditions were investigated using particle image velocimetry (PIV). The investigated parameters included: jet Reynolds number (173–979), jet angle (45°, 60°, and 90°), rotational Reynolds number (0–21,000), and Grashof number (0–300,000). The result showed that inclination angle could cause unsymmetrical flow velocities in the radial direction of the stationary disk. Also, the flow above the heated stationary disk was influenced by the vortex flow generated by the buoyancy force at lower inlet flow rate. However, the flow pattern was not affected by the jet angle at lower jet flow rate on the high-speed rotating disk. The flow pattern can be characterized by using a series of dimensionless parameters, including rotational Reynolds number, jet Reynolds number, and Grashof number.

Modelling of flutter running waves in turbine blades cascade

Flutter running waves in turbine blade cascade were often observed at experimental investigation of blades vibrations, but theirs detailed analysis has not yet been sufficiently realized. This contribution tries to explain origin and behavior of running flutter waves in the rotating blade cascade excited by steam flow from the stationary bladed disk. One of possible reasons for existence of running waves are the running periodic forces from steam wakes due to different numbers of blades of stationary and rotating wheels. Properties – velocities, directions and modes – of these forced running waves are shown using computational model of turbine disk with 10 blades. Interaction of this kind of forced excitation with aero-elastic self-excitation – flutter – causes origin of the flutter running waves. These flutter waves, modelled by the Van der Pol formula, are analysed in the paper for this simplified form of aero-elastic self-excitations.

Natural Convection Flow of Nanofluids in Squeeze Film with an Exponential Curvature

A theoretical study of the laminar squeeze flow of copper water and alumina water nanofluids between a flat circular stationary disk and a curved circular moving disk is carried out using energy integral method. The squeeze film behaviour is examined analytically and the effects of inertia and curvature on the squeeze film pressure, load carrying capacity of the fluid and temperature are analysed. Further, the problem is solved numerically for a sinusoidal motion of the upper curved disk taking an exponential form of the gap width. It is found that the copper water nanofluid is better than alumina water nanofluid for better heat transfer rates. While high inertia forces strongly influence the squeeze film behaviour, low inertia forces are favourable for temperature distribution. Further, concave nature of the upper disk gives better squeeze film characteristics than convex disk. However, convex disk is better than concave disk as far as the temperature distribution is concerned.

Computer-extended series solution for laminar flow between a fixed impermeable disk and a porous rotating disk

PurposeThe purpose of this paper is to study the laminar boundary layer flow between a stationary nonporous disk and a porous rotating disk, both being immersed in large amount of fluid.Design/methodology/approachThe governing nonlinear momentum equations in cylindrical polar coordinates together with relevant boundary conditions are reduced to a system of coupled nonlinear ordinary differential equations (NODEs) using similarity transformations. The resulting coupled NODEs are solved using computer-extended series solution and homotopy analysis method.FindingsThe analytical solutions are explicitly expressed in terms of recurrence relation for determining the universal coefficients. The nature and location of singularity which restricts the convergence of series is analyzed by using Domb–Sykes plot. Reversion of series is used for the improvement of series. The region of validity of series is extended for much larger values of Reynolds number (R), i.e. R = 6 to 15.Originality/valueThe resulting solutions are compared with earlier works in the literature and are found to be in good agreement.

The ‘spinning disk touches stationary disk’ problem revisited: an experimental approach

A popular Newtonian mechanics problem, featured in textbooks, physics olympiads and forums alike, concerns two disks with different radii and moment of inertia that rotate differently and that touch each other. Most students struggle to calculate the final angular velocity of the disks, erroneously attempting to use different conservation laws. In this paper we propose a simple experiment that should help physics teachers explain this challenging exercise in an engaging way for the students. By using a smartphone/tablet and video analysis tools, the angular velocity of both disks can easily be tracked as a function of time, clearly showing the three stages of the interaction (before touching, only one disk rotating; touching with slippage; and touching without slippage). Processing and plotting of the data in a spreadsheet immediately shows which quantities are conserved and which are not. Several extensions to the core experiment are also suggested.