Secondary Perturbation Effects in Keplerian Accretion Disks: Elliptical Instability

Origin of turbulence in cold accretion disks, particularly in 3D, which is expected to be hydrodynamic but not magnetohydrodynamic, is a big puzzle. While the flow must exhibit some turbulence in support of the transfer of mass inward and angular momentum outward, according to the linear perturbation theory it should always be stable. We demonstrate that the 3D secondary disturbance to the primarily perturbed disk which exhibits elliptical vortices into the system solves the problem. This result is essentially applicable to the outer region of accretion disks in active galactic nuclei where the gas is significantly cold and neutral in charge and the magnetic Reynolds number is smaller than 10^4.

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

The generalizations of the Onsager model for the radial boundary layer and the Carrier-Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier-Maslen models, because they are not restricted to the limit A >> 1, though they are restricted to the limit R e >> 1 and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter A = root m Omega(2)R(2)/2k(B)T). This parameter A is the ratio of the peripheral speed, Omega R, to the most probable molecular speed, root 2k(B)T/m, the Reynolds number Re = rho w Omega R(2)/mu, where m is the molecular mass, Omega and R are the rotational speed and radius of the cylinder, k(B) is the Boltzmann constant, T is the gas temperature, rho(w) is the gas density at wall, and mu is the gas viscosity. In the case of wall forcing, analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier-Maslen equation for the velocity potential. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier-Maslen equation) are determined. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used `diffuse reflection' boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a `temperature slip' (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 10 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity. The predictions of the generalized models are also significantly better than those of the original Onsager and Carrier-Maslen models, which are restricted to thin boundary layers in the limit of high stratification parameter.

Unstable m=1 modes of counter-rotating Keplerian discs

We study the linear m= 1 counter-rotating instability in a two-component, nearly Keplerian disc. Our goal is to understand these slow modes in discs orbiting massive black holes in galactic nuclei. They are of interest not only because they are of large spatial scale and can hence dominate observations but also because they can be growing modes that are readily excited by accretion events. Self-gravity being non-local, the eigenvalue problem results in a pair of coupled integral equations, which we derive for a two-component softened gravity disc. We solve this integral eigenvalue problem numerically for various values of mass fraction in the counter-rotating component. The eigenvalues are in general complex, being real only in the absence of the counter-rotating component, or imaginary when both components have identical surface density profiles. Our main results are as follows: (i) the pattern speed appears to be non-negative, with the growth (or damping) rate being larger for larger values of the pattern speed; (ii) for a given value of the pattern speed, the growth (or damping) rate increases as the mass in the counter-rotating component increases; (iii) the number of nodes of the eigenfunctions decreases with increasing pattern speed and growth rate. Observations of lopsided brightness distributions would then be dominated by modes with the least number of nodes, which also possess the largest pattern speeds and growth rates.

Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements

This paper presents a spectral finite element formulation for uniform and tapered rotating CNT embedded polymer composite beams. The exact solution to the governing differential equation of a rotating Euler-Bernoulli beam with maximum centrifugal force is used as an interpolating function for the spectral element formulation. Free vibration and wave propagation analysis is carried out using the formulated spectral element. The present study shows the substantial effect of volume fraction and L/D ratio of CNTs in a beam on the natural frequency, impulse response and wave propagation characteristics of the rotating beam. It is found that the CNTs embedded in the matrix can make the rotating beam non-dispersive in nature at higher rotation speeds. Embedded CNTs can significantly alter the dynamics of polymer-nanocomposite beams. The results are also compared with those obtained for carbon fiber reinforced laminated composite rotating beams. It is observed that CNT reinforced rotating beams are superior in performance compared to laminated composite rotating beams. © 2012 Elsevier Ltd. All rights reserved.

Unsteady Rotating Flow Over an Impulsively Rotating Infinite Disk with Axial Magnetic Field and Suction

The unsteady rotating flow of an incompressible laminar viscous electrically conducting fluid over an impulsively rotated infinite disk in the presence of magnetic field and suction is investigated. We have considered the situation where there is a steady state initially (i.e., at t = 0, the fluid is rotating with constant angular velocity over a stationary disk). Then at t > 0, the disk is suddenly rotated with a constant angular velocity either in the same direction or in opposite direction to that of the fluid rotation which causes unsteadiness in the flow field. The effect of the impulsive motion is found to be more pronounced on the tangential shear stress than on the radial shear stress. When the disk and the fluid rotate in the same direction, the tangential shear stress at the surface changes sign in a small time interval immediately after the start of the impulsive motion.

Models for the estimation of life of stator insulation in large rotating machines

Epoxy resin bonded mica splitting is the insulation of choice for machine stators. However, this system is seen to be relatively weak under time varying mechanical stress, in particular the vibration causing delamination of mica and deboning of mica from the resin matrix. The situation is accentuated under the combined action of electrical, thermal and mechanical stress. Physical and probabilistic models for failure of such systems have been proposed by one of the authors of this paper earlier. This paper presents a pragmatic accelerated failure data acquisition and analytical paradigm under multi factor coupled stress, Electrical, Thermal. The parameters of the phenomenological model so developed are estimated based on sound statistical treatment of failure data.

Stochastically driven instability in rotating shear flows

The origin of hydrodynamic turbulence in rotating shear flows is investigated, with particular emphasis on the flows whose angular velocity decreases but whose specific angular momentum increases with the increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain the observed data. Such a mismatch between the linear theory and the observations/experiments is more severe when any hydromagnetic/magnetohydrodynamic instability and then the corresponding turbulence therein is ruled out. This work explores the effect of stochastic noise on such hydrodynamic flows. We essentially concentrate on a small section of such a flow, which is nothing but a plane shear flow supplemented by the Coriolis effect. This also mimics a small section of an astrophysical accretion disc. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbation velocities and hence large energy dissipations of perturbation, which presumably generate the instability. A range of angular velocity (Omega) profiles of the background flow, starting from that of a constant specific angular momentum (lambda = Omega r(2); r being the radial coordinate) to a constant circular velocity (v(phi) = Omega r), is explored. However, all the background angular velocities exhibit identical growth and roughness exponents of their perturbations, revealing a unique universality class for the stochastically forced hydrodynamics of rotating shear flows. This work, to the best of our knowledge, is the first attempt to understand the origin of instability and turbulence in three-dimensional Rayleigh stable rotating shear flows by introducing additive noise to the underlying linearized governing equations. This has important implications to resolve the turbulence problem in astrophysical hydrodynamic flows such as accretion discs.

Determination of Isospectral Nonuniform Rotating Beams

In this paper we look for nonuniform rotating beams that are isospectral to a given uniform nonrotating beam. A rotating nonuniform beam is isospectral to the given uniform nonrotating beam if both the beams have the same spectral properties, i.e., both the beams have the same set of natural frequencies under a given boundary condition. The Barcilon-Gottlieb type transformation is proposed that converts the governing equation of a rotating beam to that of a uniform nonrotating beam. We show that there exist rotating beams isospectral to a given uniform nonrotating beam under some special conditions. The boundary conditions we consider are clamped-free and hinged-free with an elastic hinge spring. An upper bound on the rotation speed for which isospectral beams exist is proposed. The mass and stiffness distributions for these nonuniform rotating beams which are isospectral to the given uniform nonrotating beam are obtained. We use these mass and stiffness distributions in a finite element analysis to show that the obtained beams are isospectral to the given uniform nonrotating beam. A numerical example of a beam having a rectangular cross section is presented to show the application of our analysis. DOI: 10.1115/1.4006460]

Analogy Between Rotating Euler-Bernoulli and Timoshenko Beams and Stiff Strings

The governing differential equation of a rotating beam becomes the stiff-string equation if we assume uniform tension. We find the tension in the stiff string which yields the same frequency as a rotating cantilever beam with a prescribed rotating speed and identical uniform mass and stiffness. This tension varies for different modes and are found by solving a transcendental equation using bisection method. We also find the location along the rotating beam where equivalent constant tension for the stiff string acts for a given mode. Both Euler-Bernoulli and Timoshenko beams are considered for numerical results. The results provide physical insight into relation between rotating beams and stiff string which are useful for creating basis functions for approximate methods in vibration analysis of rotating beams.

Non-rotating beams isospectral to a given rotating uniform beam

In this paper, we seek to find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The Barcilon-Gottlieb transformation is used to convert the fourth order governing equation of a non-rotating beam, to a canonical fourth order eigenvalue problem. If the coefficients in this canonical equation match with the coefficients of the uniform rotating beam equation, then the non-rotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients leads to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of non-rotating beams that are isospectral to a uniform rotating beam. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. We also verify numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. Finally, the example of beams having a rectangular cross-section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the rotating beam. (c) 2012 Elsevier Ltd. All rights reserved.

Rotating beams isospectral to axially loaded nonrotating uniform beams

In this paper, the stiffness and mass per unit length distributions of a rotating beam, which is isospectral to a given uniform axially loaded nonrotating beam, are determined analytically. The Barcilon-Gottlieb transformation is extended so that it transforms the governing equation of a rotating beam into the governing equation of a uniform, axially loaded nonrotating beam. Analysis is limited to a certain class of Euler-Bernoulli cantilever beams, where the product between the stiffness and the cube of mass per unit length is a constant. The derived mass and stiffness distributions of the rotating beam are used in a finite element analysis to confirm the frequency equivalence of the given and derived beams. Examples of physically realizable beams that have a rectangular cross section are shown as a practical application of the analysis.

Stiff string approximations in Rayleigh-Ritz method for rotating beams

The governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant. The solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the Rayleighs method. Numerical experiments show better agreement with converged finite element solutions compared to polynomials. Using this as an estimate for the first mode shape, higher mode shape approximations are obtained using Gram-Schmidt orthogonalization. Estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order Rayleigh-Ritz approximation.

Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition

In this paper we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams.