Pinhas Z. Bar-Yoseph

Stability of confined swirling flow with and without vortex breakdown

A numerical investigation of steady states, their stability, onset of oscillatory instability, and slightly supercritical unsteady regimes of an axisymmetric swirling flow of a Newtonian incompressible fluid in a closed circular cylinder with a rotating lid is presented for aspect ratio (height/radius) 1 ≤ γ ≤ 3.5. Various criteria for the appearance of vortex breakdown are discussed. It is shown that vortex breakdown takes place in this system not as a result of instability but as a continuous evolution of the stationary meridional flow with increasing Reynolds number. The dependence of the critical Reynolds number Re cr and frequency of oscillations ω cr on the aspect ratio of the cylinder γ is obtained. It is found that the neutral curve Re cr (γ) and the curve ω cr (γ) consist of three successive continuous segments corresponding to different modes of the dominant perturbation. The calculated critical parameters are in good agreement with the available experimental and numerical data for γ < 3. It is shown that the onset of the oscillatory instability does not depend on the existence of a separation bubble in the subcritical steady state. By means of a weakly nonlinear analysis it is shown that the axisymmetric oscillatory instability sets in as a result of a supercritical Hopf bifurcation for each segment of the neutral curve. A weakly nonlinear asymptotic approximation of slightly supercritical flows is carried out. The results of the weakly nonlinear analysis are verified by direct numerical solution of the unsteady Navier-Stokes equation using the finite volume method. The analysis of the supercritical flow field for aspect ratio less than 1.75, for which no steady vortex breakdown is found, shows the existence of an oscillatory vortex breakdown which develops as a result of the oscillatory instability.

The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders

Flow fields within spatially periodic arrays of cylinders arranged in square and hexagonal lattices are calculated, with microscale Reynolds number ranging between zero and 200, employing a finite element numerical scheme. The terminology of an ‘‘apparent permeability’’ is introduced to establish a relationship existing between mean velocity and macroscopic pressure gradient characterized by a finite Reynolds number flow. In contrast with the low Reynolds number ‘‘true ’’ permeability, the apparent permeability is shown here to generally depend upon the direction of the applied pressure gradient, owing to nonlinearities existing within the local fluid motion. The orientation‐dependent permeabilities of both square and hexagonal monodisperse arrays are observed to diminish with increasing Reynolds number. Similar behavior is also observed for a bidisperse square array, though the apparent permeability of the latter is shown less sensitive to Darcy velocity orientation at large Reynolds numbers in compariso...

Three-dimensional instability of axisymmetric flow in a rotating lid{cylinder enclosure

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k . The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.

Stability of multiple steady states of convection in laterally heated cavities

A parametric study of multiple steady states, their stability, onset of oscillatory instability, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is presented. Cavities with four no-slip boundaries, isothermal vertical and perfectly insulated horizontal boundaries are considered. Four distinct branches of steady-state flows are found for this configuration. A complete study of stability of each branch is performed for the aspect ratio A (length/height) of the cavity varying continuously from 1 to 11 and for two fixed values of the Prandtl number: Pr = 0 and Pr = 0.015. The results are represented as stability diagrams showing the critical parameters (critical Grashof number and the frequency at the onset of the oscillatory instability) corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backwards transitions from multi-roll to single-roll states. For better comparison with the existing experimental data, an additional stability study for varying Prandtl number (0.015 ≤ Pr ≤ 0.03) and fixed value of the aspect ratio A = 4 was carried out

New B‐Spline Finite Element approach for geometrical design and mechanical analysis

In most existing CAD systems, geometrical design and mechanical analysis are operated as completely separate modules. Intensive interaction between these modules is, however, highly desired due to the iterative nature of a typical product development process. Formulating a new, unified approach to design and analysis that provides a high level of interaction is the main purpose of this research. The idea is to integrate a mechanically based geometrical design concept with the mechanical analysis module in a uniform B-Spline Finite Element (BSFE) environment. In this paper, the BSFE method is formulated and its validity and adequacy are verified for elastic linear rod and plate models. In particular, the feasibility of applying B-spline functions as base functions of the finite element method for design and analysis is demonstrated. Unique scheme attributes based on intrinsic properties of B-spline functions are investigated in detail. Method adequacy is demonstrated by comparing convergence characteristics, complexity and computational cost to the spectral element method.

The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity

The present study is devoted to the problem of onset of oscillatory instability in convective flow of an electrically conducting fluid under an externally imposed time-independent uniform magnetic field. Convection of a low-Prandtl-number fluid in a laterally heated two-dimensional horizontal cavity is considered. Fixed values of the aspect ratio (length/height=4) and Prandtl number (Pr=0.015), which are associated with the horizontal Bridgman crystal growth process and are commonly used for benchmarking purposes, are considered. The effect of a uniform magnetic field with different magnitudes and orientations on the stability of the two distinct branches (with a single-cell or a two-cell pattern) of the steady state flows is investigated. Stability diagrams showing the dependence of the critical Grashof number on the Hartmann number are presented. It is shown that a vertical magnetic field provides the strongest stabilization effect, and also that multiplicity of steady states is suppressed by the electr...

Steady states and oscillatory instability of swirling flow in a cylinder with rotating top and bottom

In this study we present a numerical investigation of steady states, onset of oscillatory instability, and slightly supercritical oscillatory states of an axisymmetric swirling flow of a Newtonian incompressible fluid in a cylinder, with independently rotating top and bottom. The first part of the study is devoted to the influence of co‐ and counter‐rotation of the bottom on the steady vortex breakdown, which takes place in the well‐known problem of flow in a cylinder with a rotating top. It is shown that weak counter‐rotation of the bottom may suppress the vortex breakdown. Stronger counter‐rotation may induce a stable steady vortex breakdown at relatively large Reynolds numbers for which a vortex breakdown does not appear in the case of the stationary bottom. Weak corotation may promote the vortex breakdown at lower Reynolds numbers than in the cylinder with the stationary bottom. Stronger corotation leads to the detachment of the recirculation zone from the axis and the formation of an additional vorte...

3D hierarchical geometric modeling and multiscale FE analysis as a base for individualized medical diagnosis of bone structure

This paper describes a new alternative for individualized mechanical analysis of bone trabecular structure. This new method closes the gap between the classic homogenization approach that is applied to macro-scale models and the modern micro-finite element method that is applied directly to micro-scale high-resolution models. The method is based on multiresolution geometrical modeling that generates intermediate structural levels. A new method for estimating multiscale material properties has also been developed to facilitate reliable and efficient mechanical analysis. What makes this method unique is that it enables direct and interactive analysis of the model at every intermediate level. Such flexibility is of principal importance in the analysis of trabecular porous structure. The method enables physicians to zoom-in dynamically and focus on the volume of interest (VOI), thus paving the way for a large class of investigations into the mechanical behavior of bone structure. This is one of the very few methods in the field of computational bio-mechanics that applies mechanical analysis adaptively on large-scale high resolution models. The proposed computational multiscale FE method can serve as an infrastructure for a future comprehensive computerized system for diagnosis of bone structures. The aim of such a system is to assist physicians in diagnosis, prognosis, drug treatment simulation and monitoring. Such a system can provide a better understanding of the disease, and hence benefit patients by providing better and more individualized treatment and high quality healthcare. In this paper, we demonstrate the feasibility of our method on a high-resolution model of vertebra L3.

The generation of streaks and hairpin vortices from a localized vortex disturbance embedded in unbounded uniform shear flow

The similarity of the coherent structures (streaks and hairpin vortices) naturally occurring in different fully developed bounded turbulent shear flows as well as in transitional flows suggests the existence of a basic mechanism responsible for the formation of these structures, under various base flow conditions. The common elements for all such flows are the shear of the base flow and the presence of a localized vortical disturbance. The objective of the present numerical study is to examine the capability of a simple model of interaction, between a localized vortical disturbance and laminar uniform unbounded shear flow, to reproduce the generation mechanism and characteristics of the coherent structures that naturally occur in turbulent bounded shear flows. The effects of the disturbance ‘localized character’ in the stream-wise and spanwise directions as well as its initial orientation relative to the base flow are investigated by using several geometries of the initial disturbance. The results demonstrate that a small-amplitude initial disturbance (linear case) eventually evolves into a streaky structure independent of its initial geometry and orientation, whereas, a large-amplitude disturbance (strongly nonlinear case) evolves into a hairpin vortex (or a packet of hairpin vortices) independent of its geometry over a wide range of the initial disturbance orientations. The main nonlinear effects are: (i) self-induced motion, which results in the movement of the vortical structure relative to the base flow and the destruction of its streamwise symmetry, and (ii) the alignment of the vortical structure with the vorticity lines. This is unlike the linear case, where there is a strong deviation of the vorticity vector from the direction of the vortical structure. Qualitatively, the disturbance evolution is sufficiently independent of its initial geometry, whereas the associated quantitative characteristics, i.e. inclination angle, centre and strength (which is governed by the transient growth mechanism), strongly depend on the disturbance geometry. The Reynolds number is found to have a negligible effect on the kinematics of the vortical structure, but does have a significant effect on its transient growth. Finally, the formation of the asymmetric hairpin vortex, due to minor spanwise asymmetries of the initial disturbance, is demonstrated.

Nonlinear Modeling and Control of a Unicycle

A unicycle system is composed of a unicycle and a rider. This system is inherently unstable, but together with a skilled rider can be autonomously controlled and stabilized. A dynamical investigation, a control design and a numerical solution of a nonlinear unicycle autonomous model are presented. The use of a nonlinear model for the control design is shown in this paper to be of great importance. A three-rigid-body physical model was selected for the dynamical study of the system. In a linearized model important physical characteristics of the unicycle system disappear (e.g. interactions between the longitudinal and lateral systems are being neglected), and therefore it is not recommended to be used for the control design. A nonlinear control law, which replaces the rider in stabilizing the model, was derived in the present work, using a nonlinear unicycle model. A simulation study shows good performance of this controller. Time spectral element methods are developed and used for integrating the nonlinear equations of motion. The approach employs the time discontinuous Galerkin method which leads to A-stable high order accurate time integration schemes.