Simulation of Taylor-Couette flow. Part 2. Numerical results for wavy-vortex flow with one travelling wave

1984 ◽  
Vol 146 ◽  
pp. 65-113 ◽  
Author(s):  
Philip S. Marcus
Keyword(s):  
Couette Flow ◽  
Vortex Flow ◽  
Travelling Waves ◽  
Maximum Amplitude ◽  
Travelling Wave ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Taylor Couette ◽  
Taylor Couette Flow

We use a numerical method that was described in Part 1 (Marcus 1984a) to solve the time-dependent Navier-Stokes equation and boundary conditions that govern Taylor-Couette flow. We compute several stable axisymmetric Taylor-vortex equilibria and several stable non-axisymmetric wavy-vortex flows that correspond to one travelling wave. For each flow we compute the energy, angular momentum, torque, wave speed, energy dissipation rate, enstrophy, and energy and enstrophy spectra. We also plot several 2-dimensional projections of the velocity field. Using the results of the numerical calculations, we conjecture that the travelling waves are a secondary instability caused by the strong radial motion in the outflow boundaries of the Taylor vortices and are not shear instabilities associated with inflection points of the azimuthal flow. We demonstrate numerically that, at the critical Reynolds number where Taylor-vortex flow becomes unstable to wavy-vortex flow, the speed of the travelling wave is equal to the azimuthal angular velocity of the fluid at the centre of the Taylor vortices. For Reynolds numbers larger than the critical value, the travelling waves have their maximum amplitude at the comoving surface, where the comoving surface is defined to be the surface of fluid that has the same azimuthal velocity as the velocity of the travelling wave. We propose a model that explains the numerically discovered fact that both Taylor-vortex flow and the one-travelling-wave flow have exponential energy spectra such that In [E(k)] ∝ k1, where k is the Fourier harmonic number in the axial direction.

Suspension Taylor–Couette flow: co-existence of stationary and travelling waves, and the characteristics of Taylor vortices and spirals

2019 ◽  
Vol 870 ◽  
pp. 901-940 ◽  
Author(s):  
Prashanth Ramesh ◽  
S. Bharadwaj ◽  
Meheboob Alam
Keyword(s):  
Reynolds Number ◽  
Velocity Field ◽  
Couette Flow ◽  
Vortex Flow ◽  
Travelling Waves ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Spiral Vortex ◽  
Taylor Couette ◽  
Taylor Couette Flow

Flow visualization and particle image velocimetry (PIV) measurements are used to unravel the pattern transition and velocity field in the Taylor–Couette flow (TCF) of neutrally buoyant non-Brownian spheres immersed in a Newtonian fluid. With increasing Reynolds number ($Re$) or the rotation rate of the inner cylinder, the bifurcation sequence in suspension TCF remains same as in its Newtonian counterpart (i.e. from the circular Couette flow (CCF) to stationary Taylor vortex flow (TVF) and then to travelling wavy Taylor vortices (WTV) with increasing $Re$) for small particle volume fractions ($\unicode[STIX]{x1D719}<0.05$). However, at $\unicode[STIX]{x1D719}\geqslant 0.05$, non-axisymmetric patterns such as (i) the spiral vortex flow (SVF) and (ii) two mixed or co-existing states of stationary (TVF, axisymmetric) and travelling (WTV or SVF, non-axisymmetric) waves, namely (iia) the ‘TVF$+$WTV’ and (iib) the ‘TVF$+$SVF’ states, are found, with the former as a primary bifurcation from CCF. While the SVF state appears both in the ramp-up and ramp-down experiments as in the work of Majji et al. (J. Fluid Mech., vol. 835, 2018, pp. 936–969), new co-existing patterns are found only during the ramp-up protocol. The secondary bifurcation TVF $\leftrightarrow$ WTV is found to be hysteretic or sub-critical for $\unicode[STIX]{x1D719}\geqslant 0.1$. In general, there is a reduction in the value of the critical Reynolds number, i.e. $Re_{c}(\unicode[STIX]{x1D719}\neq 0)<Re_{c}(\unicode[STIX]{x1D719}=0)$, for both primary and secondary transitions. The wave speeds of both travelling waves (WTV and SVF) are approximately half of the rotational velocity of the inner cylinder, with negligible dependence on $\unicode[STIX]{x1D719}$. The analysis of the radial–axial velocity field reveals that the Taylor vortices in a suspension are asymmetric and become increasingly anharmonic, with enhanced radial transport, with increasing particle loading. Instantaneous streamline patterns on the axial–radial plane confirm that the stationary Taylor vortices can indeed co-exist either with axially propagating spiral vortices or azimuthally propagating wavy Taylor vortices – their long-time stability is demonstrated. It is shown that the azimuthal velocity is considerably altered for $\unicode[STIX]{x1D719}\geqslant 0.05$, resembling shear-band type profiles, even in the CCF regime (i.e. at sub-critical Reynolds numbers) of suspension TCF; its possible role on the genesis of observed patterns as well as on the torque scaling is discussed.

Experimental Study of Axial Slit Wall Effect on Taylor-Couette Flow

2007 ◽  
Author(s):  
Sang-Hyuk Lee ◽  
Hyoung-Bum Kim
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Vortex Flow ◽  
Outer Cylinder ◽  
Velocity Fields ◽  
Taylor Vortex ◽  
Stable Mode ◽  
Taylor Vortex Flow ◽  
Taylor Couette ◽  
Taylor Couette Flow

Taylor-Couette flow has been studied extensively and lots of variables which affect the flow instability are being reported. The wall geometry effect of Taylor-Couette flow, however, has been less studied. In this study, we investigated the effect of axial slit of outer cylinder. This kind of configuration can be easily seen in rotating machinery. Particle image velocimetry method was used to measure the velocity fields in longitudinal and latitudinal planes. The index matching method was used to avoid light refraction. The velocity fields between the slit and plain model which has the smooth wall were compared. From the experiments, both models have the same flow mode below Re = 143. The transition from circular Couette flow to plain Taylor vortex flow began at Re = 103, and the next transition to wavy vortex flow occurred at 124. The effect of slit wall appeared when the Reynolds number is larger than Re = 143. Above this Reynolds number, there was no stable mode and plain and wavy Taylor vortex flow randomly appeared.

Hydromagnetic Taylor–Couette flow: wavy modes

2002 ◽  
Vol 472 ◽  
pp. 399-410 ◽  
Author(s):  
A. P. WILLIS ◽  
C. F. BARENGHI
Keyword(s):  
Magnetic Field ◽  
Couette Flow ◽  
Vortex Flow ◽  
Axial Magnetic Field ◽  
Taylor Vortex ◽  
Taylor Vortex Flow ◽  
Critical Reynolds Numbers ◽  
Taylor Couette ◽  
Axisymmetric Solutions ◽  
Taylor Couette Flow

We investigate magnetic Taylor–Couette flow in the presence of an imposed axial magnetic field. First we calculate nonlinear steady axisymmetric solutions and determine how their strength depends on the applied magnetic field. Then we perturb these solutions to find the critical Reynolds numbers for the appearance of wavy modes, and the related wave speeds, at increasing magnetic field strength. We find that values of imposed magnetic field which alter only slightly the transition from circular-Couette flow to Taylor-vortex flow, can shift the transition from Taylor-vortex flow to wavy modes by a substantial amount. The results are compared to those for onset in the absence of a magnetic field.

A spectral method for Taylor vortex flow and Taylor-Couette flow

1997 ◽  
Author(s):  
J. Rigopoulos ◽  
J. Sheridan ◽  
M. Thompson ◽  
J. Rigopoulos ◽  
J. Sheridan ◽  
...  
Keyword(s):  
Spectral Method ◽  
Couette Flow ◽  
Vortex Flow ◽  
Taylor Vortex ◽  
Taylor Vortex Flow ◽  
Taylor Couette ◽  
Taylor Couette Flow

scholarly journals Statistics, plumes and azimuthally travelling waves in ultimate Taylor–Couette turbulent vortices

2019 ◽  
Vol 876 ◽  
pp. 733-765
Author(s):  
Andreas Froitzheim ◽  
Rodrigo Ezeta ◽  
Sander G. Huisman ◽  
Sebastian Merbold ◽  
Chao Sun ◽  
...  
Keyword(s):  
Nusselt Number ◽  
Large Scale ◽  
Travelling Waves ◽  
Azimuthal Velocity ◽  
Taylor Vortex ◽  
Small Scale ◽  
Underlying Structure ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Taylor Couette

In this paper, we experimentally study the influence of large-scale Taylor rolls on the small-scale statistics and the flow organization in fully turbulent Taylor–Couette flow for Reynolds numbers up to $Re_{S}=3\times 10^{5}$. The velocity field in the gap confined by coaxial and independently rotating cylinders at a radius ratio of $\unicode[STIX]{x1D702}=0.714$ is measured using planar particle image velocimetry in horizontal planes at different cylinder heights. Flow regions with and without prominent Taylor vortices are compared. We show that the local angular momentum transport (expressed in terms of a Nusselt number) mainly takes place in the regions of the vortex in- and outflow, where the radial and azimuthal velocity components are highly correlated. The efficient momentum transfer is reflected in intermittent bursts, which becomes visible in the exponential tails of the probability density functions of the local Nusselt number. In addition, by calculating azimuthal energy co-spectra, small-scale plumes are revealed to be the underlying structure of these bursts. These flow features are very similar to the one observed in Rayleigh–Bénard convection, which emphasizes the analogies of these systems. By performing a complex proper orthogonal decomposition, we remarkably detect azimuthally travelling waves superimposed on the turbulent Taylor vortices, not only in the classical but also in the ultimate regime. This very large-scale flow pattern, which is most pronounced at the axial location of the vortex centre, is similar to the well-known wavy Taylor vortex flow, which has comparable wave speeds, but much larger azimuthal wavenumbers.

scholarly journals Effect of the number of vortices on the torque scaling in Taylor–Couette flow

2014 ◽  
Vol 748 ◽  
pp. 756-767 ◽  
Author(s):  
B. Martínez-Arias ◽  
J. Peixinho ◽  
O. Crumeyrolle ◽  
I. Mutabazi
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Couette Flow ◽  
Scaling Laws ◽  
Taylor Vortex ◽  
Large Radius ◽  
Aspect Ratios ◽  
Taylor Vortex Flow ◽  
Taylor Couette ◽  
Taylor Couette Flow

AbstractTorque measurements in Taylor–Couette flow, with large radius ratio and large aspect ratio, over a range of velocities up to a Reynolds number of 24 000 are presented. Following a specific procedure, nine states with distinct numbers of vortices along the axis were found and the aspect ratios of the vortices were measured. The relationship between the speed and the torque for a given number of vortices is reported. In the turbulent Taylor vortex flow regime, at relatively high Reynolds number, a change in behaviour is observed corresponding to intersections of the torque–speed curves for different states. Before each intersection, the torque for a state with a larger number of vortices is higher. After each intersection, the torque for a state with a larger number of vortices is lower. The exponent, from the scaling laws of the torque, always depends on the aspect ratio of the vortices. When the Reynolds number is rescaled using the mean aspect ratio of the vortices, only a partial collapse of the exponent data is found.

scholarly journals Secondary instabilities in Taylor–Couette flow of shear-thinning fluids

2021 ◽  
Vol 933 ◽  
Author(s):  
S. Topayev ◽  
C. Nouar ◽  
J. Dusek
Keyword(s):  
Numerical Simulations ◽  
Couette Flow ◽  
Vortex Flow ◽  
Shear Thinning ◽  
Taylor Vortex ◽  
Shear Thinning Fluids ◽  
Taylor Vortex Flow ◽  
Viscous Shear ◽  
Taylor Couette ◽  
Shear Thinning Fluid

The stability of the Taylor vortex flow in Newtonian and shear-thinning fluids is investigated in the case of a wide gap Taylor–Couette system. The considered radius ratio is $\eta = R_1/R_2=0.4$ . The aspect ratio (length over the gap width) of experimental configuration is 32. Flow visualization and measurements of two-dimensional flow fields with particle image velocimetry are performed in a glycerol aqueous solution (Newtonian fluid) and in xanthan gum aqueous solutions (shear-thinning fluids). The experiments are accompanied by axisymmetric numerical simulations of Taylor–Couette flow in the same gap of a Newtonian and a purely viscous shear-thinning fluid described by the Carreau model. The experimentally observed critical Reynolds and wavenumbers at the onset of Taylor vortices are in very good agreement with that obtained from a linear theory assuming a purely viscous shear-thinning fluid and infinitely long cylinders. They are not affected by the viscoelasticity of the used fluids. For the Newtonian fluid, the Taylor vortex flow (TVF) regime is found to bifurcate into a wavy vortex flow with a high frequency and low amplitude of axial oscillations of the vortices at ${Re} = 5.28 \, {Re}_c$ . At ${Re} = 6.9 \, {Re}_c$ , the frequency of oscillations decreases and the amplitude increases abruptly. For the shear-thinning fluids the secondary instability conserves axisymmetry. The latter is characterized by an instability of the array of vortices leading to a continuous sequence of creation and merging of vortex pairs. Axisymmetric numerical simulations reproduce qualitatively very well the experimentally observed flow behaviour.

Spatio-temporal character of non-wavy and wavy Taylor–Couette flow

1998 ◽  
Vol 364 ◽  
pp. 59-80 ◽  
Author(s):  
STEVEN T. WERELEY ◽  
RICHARD M. LUEPTOW
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Vortex Flow ◽  
Outer Cylinder ◽  
Taylor Vortex ◽  
Meridional Plane ◽  
Azimuthal Wave ◽  
Moderate Reynolds Number ◽  
Taylor Couette ◽  
Taylor Couette Flow

The stability of supercritical Couette flow has been studied extensively, but few measurements of the velocity field of flow have been made. Particle image velocimetry (PIV) was used to measure the axial and radial velocities in a meridional plane for non-wavy and wavy Taylor–Couette flow in the annulus between a rotating inner cylinder and a fixed outer cylinder with fixed end conditions. The experimental results for the Taylor vortex flow indicate that as the inner cylinder Reynolds number increases, the vortices become stronger and the outflow between pairs of vortices becomes increasingly jet-like. Wavy vortex flow is characterized by azimuthally wavy deformation of the vortices both axially and radially. The axial motion of the vortex centres decreases monotonically with increasing Reynolds number, but the radial motion of the vortex centres has a maximum at a moderate Reynolds number above that required for transition. Significant transfer of fluid between neighbouring vortices occurs in a cyclic fashion at certain points along an azimuthal wave, so that while one vortex grows in size, the two adjacent vortices become smaller, and vice versa. At other points in the azimuthal wave, there is an azimuthally local net axial flow in which fluid winds around the vortices with a sense corresponding to the axial deformation of the wavy vortex tube. These measurements also confirm that the shift-and-reflect symmetry used in computational studies of wavy vortex flow is a valid approach.

scholarly journals Multiplicity of states in Taylor-Couette flow of a dense granular gas

2021 ◽  
Vol 249 ◽  
pp. 03015
Author(s):  
Nandu Gopan ◽  
Meheboob Alam
Keyword(s):  
Couette Flow ◽  
Outer Cylinder ◽  
Taylor Vortex ◽  
Damping Force ◽  
Granular Gas ◽  
Taylor Vortex Flow ◽  
Wide Range ◽  
Taylor Couette ◽  
Dynamics Simulations ◽  
Taylor Couette Flow

Molecular dynamics simulations with a purely repulsive Lennard-Jones potential and a normal damping force is used to simulate the granular flow in the annular region between two differentially-rotating cylinders, called the Taylor-Couette flow. The flow transition from the azimuthally-invariant Circular Couette flow (CCF) to the Taylor-vortex flow (TVF) is studied by increasing the rotation rate (ωi) of the inner cylinder, with the outer cylinder being kept stationary. Multiplicity of states, highlighting the hysteretic nature of the “CCF ↔ TVF” transition, is observed over a wide range of rotation rates. The onset of Taylor vortices is quantified in terms of the maximum radial velocity and the net circulation per vortex.

Development and Effects of Super-Critical Taylor-Vortex Flow in a Lightly Loaded Journal Bearing

1974 ◽  
Vol 96 (1) ◽  
pp. 28-35 ◽  
Author(s):  
R. C. DiPrima ◽  
J. T. Stuart
Keyword(s):  
Vortex Flow ◽  
Axial Direction ◽  
Basic Flow ◽  
Circular Cylinders ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Secondary Motion ◽  
The Stability ◽  
Unified Description

At sufficiently high operating speeds in lightly loaded journal bearings the basic laminar flow will be unstable. The instability leads to a new steady secondary motion of ring vortices around the cylinders with a regular periodicity in the axial direction and a strength that depends on the azimuthial position (Taylor vortices). Very recently published work on the basic flow and the stability of the basic flow between eccentric circular cylinders with the inner cylinder rotating is summarized so as to provide a unified description. A procedure for calculating the Taylor-vortex flow is developed, a comparison with observed properties of the flow field is made, and formulas for the load and torque are given.

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