Taylor-Couette Problem Research Articles

Tertiary and Quaternary States in the Taylor-Couette System

The analysis of the Taylor-Couette problem in the small gap limit is extended to the cases of tertiary and quaternary solutions. The theoretical results are compared with experimental observations. Although in the latter the small-gap approximation is not always well approximated, the comparison of theoretical results and observations yields reasonable agreements. The absence of the wavy twist mode in the observed patterns is explained by the presence of no-slip boundary conditions in the axial direction of the experimental apparatus, which differ from the periodic conditions imposed in the theoretical analysis. Quaternary solutions bifurcating from the tertiary ones through subharmonic instabilities are presented and compared with experimental observations. Reasonable agreement has been found.

Low-Dimensional Analysis and Numerical Simulation of the Incompressible Flow Between Two Concentric Rotating Cylinders

There has been a very large number of experimental and theoretical studies of flow between concentric rotating cylinders in the century, since these pioneering works, the instability of Couette flow and Taylor vortex flow, now known as the “Couette–Taylor problem” is a paradigm of nonlinear problem. In order to explore the transition way of Couette–Taylor flow from laminar to turbulence and the characteristics of chaotic attractors in the turbulent regime of Couette–Taylor flow, dynamical behaviors and numerical simulation of Couette–Taylor flow have been studied by using low-dimensional model analysis method in this paper. Dynamical behaviors of a three-model Lorenz-like system of Couette–Taylor flow have been discussed, such as the stability of equilibrium, presence of limit cycles, occurrence of bifurcation and chaos, as well as the analysis of global stability etc. Moreover, using these results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment. By means of numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum and Poincare map we analyze complex dynamic behaviors of the system from the bifurcation transition to chaos.

Attractivity of the Ginzburg–Landau mode distribution for a pattern forming system with marginally stable long modes

The mathematical theory of the description of pattern forming systems close to the first instability via the Ginzburg–Landau equation is based on approximation and attractivity results. This theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the Couette–Taylor problem. Recently, approximation results for the Ginzburg–Landau approximation for pattern forming systems with marginally stable long modes, such as the Bénard–Marangoni system, have been shown. It is the purpose of this paper to prove the second fundamental property, namely the attractivity of the Ginzburg–Landau mode distribution, for such systems.

An analysis and an affordable regularization technique for the spurious force oscillations in the context of direct-forcing immersed boundary methods

The framework of this paper is the improvement of direct-forcing immersed boundary methods in presence of moving obstacles. In particular, motivations for the use of the Direct Forcing (DF) method can be found in the advantage of a fixed computational mesh for fluid–structure interaction problems. Unfortunately, the direct forcing approach suffers a serious drawback in case of moving obstacles: the well known spurious force oscillations (SFOs). In this paper, we strengthen previous analyses of the origin of the SFO through a rigorous numerical evaluation based on Taylor expansions. We propose a remedy through an easy-to-implement regularization process (regularized DF). Formally, this regularization is related to the blending of the Navier–Stokes solver with the interpolation, but no modification of the numerical scheme is needed. This approach significantly cuts off the SFOs without increasing the computational cost. The accuracy and the space convergence order of the standard DF method are conserved. This is illustrated on numerical and physical validation test cases ranging from the Taylor–Couette problem to a cylinder with an imposed sinusoidal motion subjected to a cross-flow.

Criteria and limits for flow modes of the spherical Taylor-Couette problem in medium and wide gaps

Motivated by recent theoretical and experimental work, we study the spherical Taylor-Couette flow with a view to setting qualitative criteria of the formation of the various vortex modes in relation to certain physical variables. We work on medium and large aspect ratios, namely σ = 0.154, 0.18, 0.38 and 0.48 and we are able to establish in all these cases the following criterion: Transition from one vortex mode to another occurs at those Reynolds numbers at which the torque acting on the outer sphere, the work performed on each rotation by the tangential force acting on the inner sphere and the kinetic energy of the fluid vary abruptly. We propose a physical interpretation for the conditions at which the basic vortex flow is converted into 1-vortex flow. Additionally, we determine the upper limiting value of the aspect ratio, which is σ = 0.51, beyond which Taylor vortices do not exist in annular spherical flow and we numerically obtain for the first time Taylor vortices at large aspect ratios σ = 0.48-0.51. We also show that the range (Re1 ,R e2) of the Reynolds numbers within which such vortices exist decreases with increasing aspect ratio and we point out that its span is affected by three factors. The aspect ratio σ ,t he angular velocity ωo of the outer sphere and the Grashof number Gr of the flow. The simulation of Taylor vortices in spherical flow is achieved using an efficient numerical technique and routes leading to multiple flow states are designed for large gaps. Results can be useful for the solution of such kind of boundary value physical flow problems within shell domains in several geometries.

Steady Bifurcating Solutions of the Couette–Taylor Problem for Flow in a Deformable Cylinder

The classical Couette–Taylor problem is to describe the motion of a viscous incompressible fluid in the region between two rigid coaxial cylinders, which rotate at constant angular velocities. This paper treats a generalization of this problem in which the rigid outer cylinder is replaced by a deformable (nonlinearly elastic) cylinder. The inner cylinder is rigid and rotates at a prescribed angular velocity. We study steady rotationally symmetric motions of the fluid coupled with steady axisymmetric motions of the deformable outer cylinder in which it rotates at a prescribed constant angular velocity, typically different from that of the inner cylinder. The motion of the outer cylinder is governed by a geometrically exact theory of shells and the motion of the liquid by the Navier–Stokes equations, with the domain occupied by the liquid depending on the deformation of the outer cylinder. The nonlinear fluid-solid system admits a (trivial) steady solution, termed the Couette solution, which can be found explicitly. This paper treats the global (multiparameter) bifurcation of steady-state solutions from the Couette solution. This problem exhibits technical mathematical difficulties directly due to the fluid-solid interaction: The smoothness of the shell’s configuration restricts the smoothness of the fluid variables, and their boundary values on the shell determine the smoothness of the shell’s configuration. It is essential to ensure that this cycle of implications is consistent.

The onset of resonance processes in the Couette–Taylor problem close to the point of codimension-2 bifurcation

The bifurcations on passing around the point of intersection of two neutral curves (points of codimension-2 bifurcation) are considered in the Couette–Taylor problem of the fluid motion between rotating cylinders. The secondary modes in a small neighbourhood of a point of codimension-2 bifurcation are studied using a system of non-linear amplitude equations in a central manifold. The steady-state solutions of the amplitude systems, to which secondary periodic modes of the travelling-wave type, non-linear mixtures of travelling waves and unsteady two-, three- and four-frequency quasiperiodic solutions of the system of Navier–Stokes equations correspond, are analysed. A numerical analysis of the conditions for the existence and stability of irrotationally symmetric steady-state fluid flows between unidirectionally rotating cylinders is carried out.

Heat transfer and hydraulic losses in the gap between rotating cylinders

Despite the large number of works devoted to the study of flow between coaxial rotating cylinders, little attention was given to the issues related to the complex influence of complementary factors, such as surface roughness and surface relief, the presence of the forced flow of fluid through the annular gap, on the heat transfer and flow structure. This is due to a major complication formulation of the problem when these additional factors are considered. At the same time, these factors must be considered in order to cover the existing practical applications. In this paper we examine the effect of roughness and surface relief, the forced flow of fluid through the annular gap on heat transfer between the cylinders and flow, as well as on hydraulic losses. A comparison of exiting data about impact of various factors on heat transfer and hydraulic losses is carried out on the common methodological basis. The examination is based on the known results about the impact of forced axial flow and longitudinal grooves on the cylinders surface and includes the recent data of numerical simulation of Taylor-Couette problem with radial through-flow.The results show that the radial flow through the surface of rotating cylinder provides a substantially larger increase in the heat transfer coefficient in comparison with other examined methods. The data obtained justify the new way of improvement of the technical devices that implement convective heat transfer near the surface of a rotating cylinder

Chaos generation in the Couette-Taylor problem for permeable cylinders

The methods of the bifurcation theory of codimension two, together with computer calculations, are used to investigate stationary, periodic, and quasiperiodic flows with two and three independent frequencies, as well as chaotic regimes of fluid flow between two infinite rotating permeable concentric cylinders near the intersection of the bifurcations initiating secondary stationary flow and self-oscillations with azimuthal waves.

A second-order cut-cell method for the numerical simulation of 2D flows past obstacles

We present a new second-order method, based on the MAC scheme on cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. In this approach, the solid boundary is embedded in the cartesian computational mesh. Discretizations of the viscous and convective terms are formulated in the context of finite volume methods ensuring local conservation properties of the scheme. Classical second-order centered schemes are applied in mesh cells which are sufficiently far from the obstacle. In the mesh cells cut by the obstacle, first-order approximations are proposed. The resulting linear system is nonsymmetric but the stencil remains local as in the classical MAC scheme on cartesian grids. The linear systems are solved by a fast direct method based on the capacitance matrix method. The time integration is achieved with a second-order projection scheme. While in cut-cells the scheme is locally first-order, a global second-order accuracy is recovered. This property is assessed by computing analytical solutions for a Taylor–Couette problem. The efficiency and robustness of the method is supported by numerical simulations of 2D flows past a circular cylinder at Reynolds number up to 9500. Good agreement with experimental and published numerical results are obtained.

Transient Growth of Perturbation Energy in the Taylor—Couette Problem with Radial Flow

Transient growth of perturbation energy in the Taylor—Couette problem with radial flow is investigated. The effects of radial flow on transient growth and structure of the optimal perturbation are mainly considered. For the wide gap case, strong radial flow, either inward or outward, shifts the peak of the amplitude of optimal perturbation towards the outer cylinder and the lift-up mechanism cannot be observed. However, for the narrow gap case, the optimal perturbation is almost unaffected by the radial flow and the lift-up mechanism still exists.

Nonresonant case of intersection of bifurcation curves in the Couette-Taylor problem

The nonresonant case (Res 0) of the motion of a viscous incompressible fluid between rotating coaxial cylinders in a small neighborhood of a bifurcation point of codimension 2 is considered, where the amplitude system has only essential resonant terms. Existence and stability conditions are obtained for its solutions which correspond to various periodic and quasiperiodic solutions of the Navier-Stokes equations. In a small neighborhood of some points of the resonance Res 0, the regions of existence and stability of these solutions are determined.

Taylor–Couette flow of ferrofluid: Spin field and spin boundary condition effects

We solve for the steady flow solutions of a ferrofluid between concentric cylinders (the Taylor-Couette problem) and consider the effects of the spin field and the spin boundary conditions on the flow. In particular, our model includes the full spin equations. We analyze families of solutions for a range of realistic flow parameters and radial magnetic fields. Particular attention is paid to regimes in which different spin boundary conditions lead to significantly different flow profiles.

Resonance regimes in the neighborhood of bifurcation points of codimension 2 in the Couette-Taylor problem

The bifurcation in a dynamical system with cylindrical symmetry dependent on several parameters is studied with reference to the Couette-Taylor problem. Points at which two neutral curves intersect (bifurcation points of codimension 2) corresponding to several independent neutral modes are found. In the neighborhood of the bifurcation points of codimension 2 the interaction of these modes can be described by a system of amplitude equations on the central manifold. If the neutral modes are nonrotationally symmetrical, there exist seven different resonance states that influence the cubic terms of the amplitude system. For the resonances Res 0 and Res 3 the results of calculating the intersection points are presented and the conditions under which stationary regimes exist and are stable are analyzed.

Marginal stability analysis on salt-fingers convection with parabolic temperature and salinity profiles

This paper explores the influence of parabolic profiles of temperature and salinity, as might arise due to local evaporation or warming disturbance, on the marginal stability problem in a salt-fingering regime. From the results of this article, the governing equations obtained for stationary onset of salt-fingers convection of a gravity gradient due to local disturbance in parabolic temperature and salinity profiles are similar to those obtained by the small-gap Taylor–Couette problem. Detailed relationships between the effective Rayleigh number ( R ˜ c ), the critical wave number ( a c ) and the couple disturbed local depth under the stationary stability analysis are also shown in this work. From the result of overstability analysis, it shows that oscillatory motion of salt-fingers convection will be triggered under a certain definite characteristic frequency. In overstability, it provoke restoring forces so strong as to overshoot the corresponding position on the other side of equilibrium for case of coupled disturbance in the upper layer of ocean.

Overstability analysis on salt-fingers convection with parabolic temperature and salinity profiles

This paper explores the influence of parabolic profiles of temperature and salinity, as might arise due to local evaporation or warming disturbance, on the marginal stability problem in a salt-fingering regime. From the results of this article, the governing equations obtained for stationary onset of salt-fingers convection of a gravity gradient due to local disturbance in parabolic temperature and salinity profiles are similar to those obtained by the small-gap Taylor–Couette problem. Detailed relationships between the effective Rayleigh number ( R ˜ c ) , the critical wave number ( a c ) and the couple disturbed local depth under the stationary stability analysis are also shown in this work. The result of overstability analysis shows that the oscillatory motion of salt-fingers convection will be triggered under a certain definite characteristic frequency. In overstability, it provoke restoring forces so strong as to overshoot the corresponding position on the other side of equilibrium for case of coupled disturbance in the upper layer of ocean.

Resonances in the Codimension-2 Bifurcations in the Couette–Taylor Problem

The investigation of codimension-2 bifurcations, in particular in systems with cylindric symmetry, enables us to deduce new types of secondary regimes branching-off from the symmetric regimes. This investigation also allows us the unique possibility of a rigorous treatment of chaotic solutions to Navier–Stokes and other nonlinear PDE’s. The central manifold approach combined with the reduction to the normal form lead to the so-called amplitude systems. These ODE systems describe the nonlinear interaction between the neutral modes, and always include several nonlinear terms due to so-called intrinsic resonances. However, sometimes additional resonances appear. In this paper we present the complete list of all possible resonances in dynamic systems with cylindric symmetry and the corresponding forms of the amplitude equations. Further, we present the results of extensive numerical investigation of the resonant codimension-2 bifurcations in the Couette–Taylor problem, thus creating an intriguing subject for further investigation.

A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence

This paper is the second part of a two-part paper treating a non-self-adjoint quadratic eigenvalue problem for the linear stability of solutions to the Taylor-Couette problem for flow of a viscous liquid in a deformable cylinder, with the cylinder modelled as a membrane. The first part formulated the problem, analyzed it, and presented computations. In this second part, we first give a weak formulation of the problem, carefully contrived so that the pressure boundary terms are eliminated from the equations. We prove that the bilinear forms appearing in the weak formulation satisfy continuous inf-sup conditions. We combine a Fourier expansion with the finite element method to produce a discrete problem satisfying discrete inf-sup conditions. Finally, the Galerkin approximation theory for polynomial eigenvalue problems is applied to prove convergence of the spectrum.

A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations

This two-part paper treats the numerical approximation of a tricky quadratic eigenvalue problem arising from the following generalization of the classical Taylor-Couette problem: A viscous incompressible fluid occupies the region between a rigid inner cylinder and a deformable outer cylinder, which we take to be a nonlinearly viscoelastic membrane. The inner cylinder rotates at a prescribed angular velocity ω, driving the fluid, which in turn drives the deformable outer cylinder. The motion of the outer cylinder is not prescribed, but responds to the forces exerted on it by the moving fluid. A steady solution of this coupled fluid-solid system, analogous to the Couette solution of the classical problem, can be found analytically. Its linearized stability is governed by a non-self-adjoint quadratic eigenvalue problem. In Part I, we give a careful formulation of the geometrically exact problem. We compute the eigenvalue trajectories in the complex plane as functions of ω by using a Fourier-finite element method. Computational results show that the steady solution loses its stability by a process suggestive of a Takens-Bogdanov bifurcation. In Part II we prove convergence of the numerical method.

Adapting simulation codes to compute unstable solutions and bifurcation behaviour

AbstractSimulation codes for solving large systems of ordinary differential equations suffer from the disadvantage that bifurcation‐theoretic results about the underlying dynamical system cannot be obtained from them easily, if at all. Bifurcation behaviour typically can be inferred only after significant computational effort, and even then the exact location and nature of the bifurcation cannot always be determined definitively. By incorporating relatively minor changes to an existing simulation code for the Taylor–Couette problem, specifically, by implementing the Newton–Picard method, we have developed a computational structure that enables us to overcome some of the inherent limitations of the simulation code and begin to perform bifurcation‐theoretic tasks. While a complete bifurcation picture was not developed, three distinct solution branches of the Taylor–Couette problem were analysed. These branches exhibit a wide variety of behaviours, including Hopf bifurcation points, symmetry‐breaking bifurcation points, turning points and bifurcation to motion on a torus. Unstable equilibrium and time‐periodic solutions were also computed. Copyright © 2007 John Wiley & Sons, Ltd.